And even if you could(1), the background process typically has the terminal open...

emmelaich on Dec 8, 2014 | parent | context | favorite | on: Reptyr – Reparent a running program to a new termi...

And even if you could(1), the background process typically has the terminal open - and screen uses a separate pseudo terminal aka pty -- As suggested by the name - rePTYr

(1) ^Z sends SIGSTOP; to continue you can signal SIGCONT. But then what?!