Spurs’ striker signed a new contract at White Hart Lane in December, committing his future until 2022 on a deal worth more than £100,000 a week.
The 23-year-old England international is now the top earner at Tottenham, having become one of the Premier League’s top hitmen.
Jose Mourinho has made Atletico Madrid’s Griezmann his No 1 target come the end of the season.
But according to ESPN, Kane is seen as United’s ideal summer acquisition up front.
Manchester United's ideal two striker signings this summer would be Harry Kane and Antoine Griezmann
With Kane having shown his desire to stay long-term, it is not believed he will push for a move to Manchester United
However, senior figures at Old Trafford are said to have no expectation that Spurs will allow the proven goalscorer to leave.
And with Kane having shown his desire to stay long-term, it is not believed he will push for a move to Manchester United either.
Mourinho apparently has concerns over his current strikers, with Zlatan Ibrahimovic his only prolific marksman.
But with the experienced Swede turning 36 in October, the Portuguese boss knows he cannot rely on him next term.
Jose Mourinho accepts Manchester United striker Zlatan Ibrahimovic is nearing the end of his career
How Man United could look with Antoine Griezmann
Wed, February 1, 2017
Is this how Manchester United's squad could look with Antoine Griezmann
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Is this how Manchester United could look with Antoine Griezmann
Jose Mourinho wants to bring in two forwards as he is worried about his options at Manchester United
Young stars Marcus Rashford and Anthony Martial have struggled to replicate their form of last season, while Wayne Rooney has also underperformed, and Mourinho has made bringing in two goalscorers in their prime a priority.
United scouts have subsequently been on the search for someone to accompany Griezmann this summer, with Mourinho having succeeded in finding forwards during his managerial career such as Diego Costa, Didier Drogba and Diego Milito.
Express Sport understands the Red Devils have not yet agreed terms with France star Griezmann, but he is a top target who the club have not denied they are desperate to sign.