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To Answer your question, the employer can file the Petition on your behalf, but in order to get approval from the USCIS, they require that your stay in the US is legal at all times and that you are not "out of status". So your first step should be to find out why there is a delay in approving your application to extend your stay in the US, so that your status in the US is, at all times "legal" and that the USCIS does not find that you are "out of status" at any time during your stay in the US.
In order for you to legally work for the employer, the employer will have to help you get a US Labor Certification so that you can legally remain and work in the US. But, in order for the USCIS to issue this Labor Certification, the USCIS requires that you remain in legal status in the US because the USCIS will not issue a Labor Certification to any person who does not have legal status in the US.
Therefore, the first thing that you must do is to contact the USCIS and ask them the reason for the delay in approving your I-539 Application to extend your stay in the US. It is very, very important that you determine the reason for the delay and to ask if there is anything that they require of you in order to approve your application. The USCIS requires that your stay in the US remains legal at all times. If the USCIS finds that you are out of status, it will not approve your employer's Petition they will not approve your request to work in the US.
If you have any questions or ned further clarification, please let me know and I will be glad to explain further.
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