Is it possible to accept a work offer while I am an intern/trainee in the U.S.?

No. The J-1 visa is only valid for working at the host company stated in your initial application.

Accepting other work offers would infringe upon U.S. visa regulations. Immediate departure to your home country will be executed. The U.S. Department of State may even ban you from coming back to the U.S. for many years.