Can I start perm when an H-1B transfer is under process?
Yes, your new employer can start the PERM for you at any time.
Remember that PERM and thereby Green card application is for FUTURE employment. Therefore, there is no need to worry with H1B transfer.
Even if your H1B transfer is denied, your new employer’s Green Card (PERM application) will still remain valid and will continue its due course of decision making.
You don’t even have to be in the United States in order for a company to file a PERM or I-140 on your behalf.