The 21-year-old was in impressive form for Forest during the 2021-22 campaign, playing a key role in their promotion back to the Premier League.
Johnson scored 16 goals and registered nine assists in 46 appearances during the regular season before netting twice in three playoff fixtures, registering in both legs of their semi-final against Sheffield United.
The attacker only has little over 12 months left to run on his current deal, which has led to speculation surrounding his future with Steve Cooper's side.
According to The United Stand, Johnson has emerged as an option for the 20-time English champions, who are keen to improve their attacking option for the 2022-23 season.
The report claims that the forward was discussed when Man United held talks with Forest over a possible loan deal for English goalkeeper Dean Henderson.
The Reds are believed to value Johnson between £30m and £40m but are not interested in selling him ahead of their long-awaited return to the top flight of English football.
However, Man United could potentially attempt to use their strong relationship with Forest to sign the Wales international at a later date.
Johnson has found the back of the net twice in 13 appearances for his country, netting in their last two fixtures in the UEFA Nations League against Belgium and the Netherlands.
The forward is now in a strong position to secure a spot in the Wales squad for the 2022 World Cup, with Dragons boss Rob Page recently lavishing price on the former Dunkirk youngster.
"He's a wonderful talent. I'm looking forward to watching him in the Premier League next season and watching him grow with us because every time he comes on he impresses, shows his worth so he deserved his goal. He's got a lot more to come as well," Page told reporters.
Man United are allegedly currently in talks with Barcelona over Frenkie de Jong, while Porto's Vitinha and Derby County's Malcolm Ebiowei are said to be wanted by the Red Devils.
However, the club are reportedly facing a battle to keep hold of Cristiano Ronaldo, with the Portugal international said to be eyeing a move away from Old Trafford this summer.
Man United will begin their pre-season against Liverpool on July 12 before also taking on Melbourne City, Crystal Palace, Aston Villa, Atletico Madrid and Rayo Vallecano as part of their preparations for the new campaign.