City are among a number of clubs who have registered an interest in Pogba, who is set to leave Old Trafford as a free agent in June.
PSG are leading the race, in part because Juventus are not in a position to match the French club's financial offer.
City are aware they are not top of Pogba's list but sources have told ESPN that his contract situation has made the 29-year-old an attractive prospect ahead of a summer during which they are exploring options to bolster their midfield.
An official announcement could come as early as this week.
Sources have told ESPN that Pogba will not make a decision about his own future until the end of the season.
United have one game remaining against Crystal Palace on May 22 although the World Cup winner is facing a race against time to be fit as he continues his recovery from a muscle injury.
Meanwhile, United are close to confirming Mitchell van der Gaag as one of Erik ten Hag's backroom staff once he takes over at Old Trafford in the summer.
Van der Gaag has been working with Ten Hag at Ajax and is set to follow the 52-year-old to United.
Ten Hag also approached Fred Rutten to join his coaching team but sources have told ESPN he is set to turn down the offer in favour of taking a job at PSV Eindhoven.