If he's not on a legal status such as H1-B or F-1, the employer can NOT spomsor him for green card. The reason is that they will eventually have to file for "Adjust of Status to GC" and since the applicant has no status, you can't adjust it to GC. They will have to leave the US and request consular processing of an immigrant visa. Departing the US will automatically nullify his pending asylum application and there won't be any guarantee if the US embassy abroad will him the immigrant visa or not. However, if your friend has maintained his status (has been working through OPT or H1-B), he can get an employent based green card without leaving the US. Remember that a pending asylum is not a legal status, it's a lawful presence.
If he has no legal status, I don't recommend abandoning the asylum. Just keep working for the same company and ask them to file for labor certificate and I-140 while the asylum is pending. if I-140 approved, you cannot file I-485 to get GC because you have no status. instead, wait till you get your final asylum denial and then request consular processing of an immigrant visa through the employer -> leave the US -> re-renter with the visa -> get your GC