The 26-year-old impressed for the Bees during the 2021-22 campaign, finding the back of the net on 14 occasions in 37 appearances in all competitions, in addition to contributing six assists.
Toney scored 12 times and registered five assists in the Premier League, as Thomas Frank's side finished 13th in England's top flight during an impressive season.
According to The Mirror, Man United are admirers of the striker and could make a move for him before the summer transfer window closes for business.
The report claims that Newcastle United, West Ham United and Leeds United are also keen on Toney, who still has three years left to run on his current deal.
Toney scored 49 goals in 94 appearances for Peterborough United before making the move to Brentford, and he helped fire the Bees into the Premier League with 31 Championship goals in 2020-21.
Frank's side are allegedly expecting 'at least one bid' for the striker before the end of the window.
Newcastle, West Ham and Leeds are all keen to bring a new forward to the club this summer, while Man United are also light in the final third of the field following Edinson Cavani's departure on a free transfer.
Anthony Martial is back at Old Trafford, and the France international has impressed in the early stages of pre-season, scoring twice in the victories over Liverpool and Melbourne Victory.
However, there remains uncertainty surrounding Cristiano Ronaldo's future at the club, with the Portugal international pushing to leave before the end of the transfer window.
Paris Saint-Germain and Chelsea have both recently ruled themselves out of the running for the 37-year-old, though, while Bayern Munich have reiterated that they are not interested in his services despite agreeing to sell Robert Lewandowski to Barcelona.
"I have a lot of respect for Cristiano Ronaldo, his successes and his career. But once again: That was and is not a topic for us," Bayern sporting director Hasan Salihamidzic told SPORT1.
Mason Greenwood remains suspended by the Red Devils, further limiting the options of new head coach Erik ten Hag when it comes to the centre-forward spot at Old Trafford.
Man United confirmed Christian Eriksen as their second signing of the summer on Friday, with the Denmark international following Tyrell Malacia to the club.
The 20-time English champions are believed to be on the verge of signing Lisandro Martinez from Ajax, but a deal for Barcelona's Frenkie de Jong continues to stall due to the midfielder's desire to remain at Camp Nou.