The Sun says United are confident of announcing the news at the end of the season.
The Frenchman, 27, has had a change of heart since expressing his desire to leave the club this time last season.
Boss Ole Gunnar Solskjaer has reinstated him into his post-lockdown team after he recovered from an ankle injury.
While Pogba looked tired and made a couple of errors against Southampton — one of which led to their first goal in Monday's 2-2 draw — he has generally been playing well.
Solskjaer had started to plan for the future without him but has noticed a real change in attitude and form.