No, it is not possible to receive a green card with a name other than your actual official name.
Once you're a permanent resident in the US, you can follow the same process for name change as everyone else (usually involves State court signing off on a name change petition), and then applying for a green card replacement due to the name change. You should also check with your country of citizenship, though, since their requirements may differ and may require you to use that country's process for name change.
If you're going through a naturalization in the US, you can ask for a name change as part of the process. In that case, your naturalization ceremony will be held in front of a Federal judge who'd also sign off on the name change as part of the ceremony and you'll get your naturalization certificate with the new name (this may delay the process a bit though, since not every naturalization ceremony is scheduled to be in front of a Federal judge).