The Red Devils were heavily linked with a move to sign the France international in the summer transfer window.
However, Griezmann opted to commit his future to Atletico Madrid following los Rojiblancos’ transfer ban.
The France international has continued to be linked with a move to Manchester United but Barcelona have entered the race.
Griezmann has scored five goals and has made three assists in 15 games in La Liga this term to make a slow start.
However, Barcelona are thought to be ready to step up their interest in Griezmann ahead of the January transfer window.
New York Times reporter Smith claimed that Manchester United could miss out due to bad timing.
“Man United would like him next summer,” Smith told BBC Radio 5 Live.
“If Barcelona move in January and Manchester United aren’t in a position, they’ve been undone by the timing.”
Manchester United ended up signing Everton striker Romelu Lukaku in a £75m deal in the summer transfer window.
Griezmann’s close France pal Paul Pogba is currently at Manchester United.
Manchester United are looking to win the Premier League title for the first time since 2013 when Sir Alex Ferguson was still in charge.
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