I always made sure the company will sponsor for綠卡

before I took the offer. However, if you only submitted this request after started working already. I really think it depends on your performance & relationship with your direct supervisor. If the company sponsored GC for employees before, I don't see any reason they would not do the same for you giving your performance is satisfactory & you are a fit not only for the position, but also for the company as well.

If you are sure you have a strong status there, tell your direct supervisor about your need sincerely & explain fully why it is a necessaty for you to continue to work for the company; If you are unsure about how your direct boss would evaluate your work, go get a back up offer first before you talk to your direct boss about it. If it does not work with this company, you can still go for your new offer.

Good luck.