The experienced duo only have deals with the 20-time English champions until the end of June, and it had been thought that both would depart on free transfers ahead of the 2024-25 campaign.
However, the Red Devils have revealed that talks are ongoing with the pair over the possibility of extending their time at Old Trafford, potentially for one more season each.
Evans joined Man United for the start of pre-season training last summer, and he was ultimately offered a one-year contract due to the impression he made both on and off the field.
The 36-year-old was only regarded as cover for Lisandro Martinez, Raphael Varane, Harry Maguire and Victor Lindelof, but he ended up making 30 appearances during the 2023-24 campaign due to the team's injury problems in the middle of their defence.
Man United in talks with Evans, Heaton over new deals
Evans featured in 23 of Man United's 38 Premier League matches last season, while he also played four times in the FA Cup, including as a substitute in the final of the competition against Manchester City.
Raphael Varane's departure on a free transfer has already been confirmed, while there is speculation surrounding Victor Lindelof's future.
As a result, Evans could seemingly stay to form part of the centre-back group next season alongside Harry Maguire, Lisandro Martinez and Willy Kambwala, with a new player also set to be signed in that area.
Evans previously represented the 20-time English champions between 2006 and 2015, winning three Premier League titles, two EFL Cups and the Champions League.
Heaton, meanwhile, is also in his second spell at Old Trafford, having initially represented the club between 2005 and 2010, although he failed to make a first-team appearance in that time.
Heaton could remain at club as number three goalkeeper
The 38-year-old returned to Man United in the summer of 2021 following his release from Aston Villa, but he has only made three appearances over the last three seasons.
Heaton featured once in 2021-22 before playing twice in 2022-23, but he did not make a single outing for the club last term, with Altay Bayindir the number two goalkeeper to Andre Onana.
The Englishman was only on the bench on four occasions in the Premier League last season, while he was left out of the squad for the FA Cup final against Manchester City.
However, Heaton could seemingly be offered the chance to remain as number three for next term, while a coaching role in the future is also thought to be a possibility.
Man United also confirmed on Wednesday afternoon that Omari Forson has been offered a new contract, but academy graduate Brandon Williams is being released.