Zaha was Alex Ferguson’s final signing as United manager in 2013, but he only made four first-team appearances for the club.
The Palace academy product returned to his boyhood side the following season, and has remained there ever since.
The Ivory Coast international handed in a transfer request in the summer, as Everton attempted to prise him away from Selhurst Park.
Arsenal were also heavily linked with the forward, but the Gunners ultimately opted to sign his international team-mate Nicolas Pepe instead.
Palace were therefore able to keep hold of their prized asset after neither of his suitors met their asking price, but they are expected to receive more offers next summer.
And according to The Sun, United could make a bid for their ex-player at the end of the current campaign, with Zaha likely to again push for a big-money move.
The club are confident that Zaha, who celebrated his 27th birthday on Sunday, is now mature enough to deliver at Old Trafford.
He had not long turned 20 when United secured his signature midway through the 2012/13 season, and struggled to adapt to life away from London.
Ole Gunnar Solskjaer has changed the Red Devils’ recruitment policy since taking charge, with the focus now on younger, hungry players with experience of English football.
Zaha fits the bill in that regard and would add another option to the team’s forward line.
Palace will not let their star man go on the cheap, however, with the south London outfit likely to demand at least £70m.
United face Brighton on Sunday knowing that victory could see them climb as high as seventh in the Premier League table if other results also go their way.