Here's the information you need: http://www.servicecanada.gc.ca/eng/ei/digest/10_2_0.shtml#a10_2_0
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10.2.4.3 Restricted to one employer
A person whose work permit includes a restriction that only allows them to work for a specific employer, is not normally considered to be available to accept work, and may be disentitled from collecting benefits.
However, the simple fact that the work permit restricts the worker to one employer is not the only factor to be considered when determining the claimant's availability.
It is important to fact find and take all factors into consideration before determining that a foreign worker is unavailable because their work permit restricts their employment to one employer.
Before making a determination, it is necessary to obtain a declaration from the claimant regarding their availability. In addition, the claimant must show that once they receive an offer of employment from a new employer, CIC will remove the restriction on their work permit.
10.2.4.4 Expired work permit
The availability of a claimant who holds a valid work permit, must be assessed based on their individual circumstances, taking into consideration all the terms of their work permit. When a claimant indicates they are available for work, and there is no issue with the work permit, or any contradictory evidence on file concerning the claimant's availability, entitlement to benefits must be considered the same as for any other claimant.
Once the work permit has expired, if the claimant cannot show that they applied for an extension prior to the expiry date, the claimant no longer has any status in Canada. In these situations, the claimant cannot prove they are available for work and a disentitlement is warranted. If the claimant proves they applied for a new work permit prior to the expiry of the previous one, availability would be considered the same as for any other claimant.
10.2.5 Subjective considerations
A person's intentions may be determined by words and actions. Actions often speak louder than words, and they should certainly be given more weight.
In fact, a person who is not really looking for work will not always say so. Unwillingness to work is sometimes disguised by undue restrictions which make it very difficult to find work and is characterized by a passive attitude toward one's responsibility in looking for work. The circumstances surrounding a voluntary separation from employment and a failure to avail oneself of a new job opportunity are other indications of non-availability where the reasons given appear at best to be a mere excuse for not working.
This type of situation resolves itself into a question of credibility and must be decided on the balance of probabilities. Where the claimant's actions are not such as to constitute evidence of availability and the statements made lack credibility, a disentitlement may be imposed since, under the legislation, the onus of proving availability lies with the claimant. Proof beyond any reasonable doubt is not required but rather the evidence must lead the adjudicating authorities to believe that the claimant is available rather than tend to indicate the contrary.
It is erroneous to believe that a disentitlement to benefits may not be imposed until the claimant has turned down a suitable job offer. On the other hand, the absence of a job refusal or good cause for not accepting in the case of a refusal may tip the scales in the claimant's favour. What is required is evidence that the claimant sincerely endeavours to put an end to the period of unemployment as quickly as possible and that the labour market situation is unfavourable despite one's willingness to work, ability and skills. The best evidence that can be provided to this end, and perhaps the only form of evidence, is a detailed account of one's personal efforts made in all areas of the labour market in which there is a reasonable hope of obtaining employment. There are no legislative provisions which govern the number and type of applications that must be made or the format for a claimant's record of job searches. (CUB's 13125, 14576, 15006, 17033) The legislation only requires that a claimant prove that they are making reasonable and customary efforts to obtain suitable employment5. Availability must be examined objectively in light of the claimant's intentions, as shown by their efforts and in light of the possibilities of employment in the kind of work sought and the region where they reside (CUB 15006).
10.2.6 Objective considerations
Canadian labour market information is available through a wide variety of internet and other media sources and details on employment opportunities can be obtained quite easily. Where the required information has not been obtained, a reasonable doubt as to employment opportunities exists, and this would remove support for a disentitlement provided the claimant's expectations do not appear unreasonable. The argument that the employment opportunities are limited to some extent since the claimant has made specific demands is not enough to support a finding that they are unduly limited.
Where the labour market information regarding employment opportunities is inadequate but the claimant's expectations appear very restrictive in nature, although they might not be qualified as extreme restrictions, the claimant should be advised that they must seek and accept employment that is consistent with the legislative definition of suitable employment, including the requirement to seek and accept less favourable earnings and occupations, as the benefit period progresses. If the claimant disregards this advice and chooses to neither seek nor accept such work, and assuming there is such suitable employment available for the claimant in the labour market, the claimant is subject to a disentitlement for non-availability.
It is erroneous to believe that in order to prove availability from then on, such a claimant must find a job. Availability may be proven afterwards by removing one's restrictions.
Finally, in cases where the demands appear to be unduly restrictive or cast doubt on one's interest in seeking employment and desire to find work as quickly as possible, a finding that availability has not been proven may be proper, even in the absence of detailed labour market information. It will be up to the claimant to remove those limitations.
The above principles also apply to a claimant whose restrictions arise out of a real constraint, such as personal or family obligations or lack of transportation, rather than mere personal demands. Should the restrictions thus imposed be so serious as to make the employment opportunities virtually non-existent, then it will be up to the individual, in order to become eligible for benefits, to find a solution that will make it possible to accept most of the employment opportunities that may arise."
From the above:
1. The fact that your work permit restricts you to one employer is not the only factor to be considered when they determine your availability for work.
You can refer them to this point and say that you think there are other factors that need to be taken into consideration such as a declaration regarding your availability to accept work and that once you receive an offer of employment from a new employer, CIC will remove the restriction. This brings us to #2 below.
2. They have to fact find and take all factors into consideration. It is necessary for them to ask you to provide a declaration regarding your availability.
It is good to provide a declaration regarding your availability for work. You can say that a restricted work permit does not affect your availability to accept work for these reasons:
a). You are or have been actively applying for jobs and attending interviews. Only say this if it's true and attach proof of your job search efforts such as dates you applied, name of the companies/employers you have sent applications to, any emails/confirmations you would have received after your Resume/CV was received, interviews scheduled/attended, etc).
b). Once you are offered employment by a new employer, the proper thing to do is to apply for a new work permit from CIC. So you are in fact available to accept work, and following the rules(which is what the government expects of applicants) by applying for a work permit. Besides, a new employer who has gone through the necessary process to offer you employment knows that it involves applying for a work permit from CIC. To do otherwise would be breaking the law(i.e. working without proper authorization) and could lead to consequences for yourself and the employer.
c). Proof of legal status is one requirement for applying for a new work permit. Your current work permit gives you legal status and can only be helpful when you apply for a new work permit. Print and attach the document checklist for applying for work permits inside Canada.
3. Also, you have to show that once you receive an offer of employment from a new employer, CIC will remove the restriction.
You can explain that a job offer from the new employer along with a positive Labor Market Opinion will enable you obtain a new work permit with the new employer's name on it. This in itself removes the restriction because:
a) the new employer's name on your new work permit allows you to work for that employer
b) the work permit with the old employer's name on it is no longer needed
c) you can renew the work permit
I hope this is helpful. Remember to be polite and tactful. haha
On a different note, I have read about people with AOR applying for BOWP eventhough their current "closed" work permit are not expiring within 4 months and they were successful. They wrote an explanation letter. Perhaps you could consider this?