Call me dumb, h1 expiring...

This is my situation:

My h-1 started in 1999 and will expire in Dec this year, I never had the chance to start green card process. I was just let go from my job last month and looking for a new one.

Currently I am in the 3rd round of an interview for a job. This is a very good opportunity I hate to pass up, but with only 10 months left on my H-1, how can I convince the company to hire me? Is there enough time for the company to start my green card, say, through PERM?

Thanks a lot.

Zoe