Unfortunately, it's not possible to transfer a contract to a different account. Your company can hire your freelancer on a new contract and then you can end the contract you have with him through your account.
Thanks, I was told by him not to do that because he had to do that before and his account got flagged as fraudulent because the upwork automated system thought he was hiring himself. I suppose because, just as in my case, the company had never hired someone on upwork before.
Soo....? What am I supposed to do?
Will you be managing your company's client account?
Generally, there is nothing wrong with you sending a link to the freelancer's profile to another client so they could hire them via their account. It's recommended that communication between the freelancer and the client are kept on Upwork or at least main points are recorded in Upwork Messages in case any issues arise.