approved visa for Old Passport to be stamped on New Passport

If I get a new passport can the approved visa be stamped on my new passport?

You cannot receive a U.S. visa stamp in the U.S., even though with the appropriate permissions you may stay legally, change your status (e.g. from visitor to worker) and start employment.

You do not need a visa to stay in the U.S. but you will have to have one if you want to visit other countries and re-enter the U.S. (in most cases).

When you actually apply for a visa sticker outside of the U.S. (using the DS-160 form), you can simply put your new passport number and use your new passport to receive your visa sticker based on your approved petition (which is issued for you as a person).

If you already submitted DS-160 or it takes a long time to renew the passport, you could try asking the consular officer during your interview to make the change (bring both passports!).

Note that the U.S. accepts valid visa stickers in expired passports; so if you cannot have your new passport in time, the old one may still be an option (if it has not expired when the visa is issued). But of course, you will need to travel with the two passports every time you want to enter the U.S. which may be a bit inconvenient.