The Manchester Evening News claimed the France international is set to complete an £86million move to Old Trafford at the end of the season.
However, The Independent report that United still have a long way to go before they land their man.
Club chiefs fear Griezmann could simply be using their interest in order to force Real Madrid to come in for him.
And it is said they have considered three alternatives if they miss out on the 26-year-old.
How Man United could look with Antoine Griezmann
Thu, March 23, 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
Manchester United are considering Antoine Griezmann alternatives
Robert Lewandowski is seen as unattainable by Manchester United chiefs
The first is Bayern Munich star Robert Lewandowski who has scored 33 goals from 37 games for his club this season.
The Poland international nearly arrived at United in 2012 but the Red Devils opted for Arsenal hitman Robin van Persie instead.
Despite liking Lewandowski, however, United fear he would be too difficult to obtain.
Other options include Everton's Romelu Lukaku and Monaco's Kylian Mbappe.
Kylian Mbappe is wanted by Manchester United and Real Madrid
Lukaku looks set to leave Everton after refusing to sign a new contract.
United could attempt to sign him but Mourinho is reluctant to meet the Toffees' £70million asking price.
Mbappe, meanwhile, is on United's radar but has also been linked with a move to Real Madrid.