The France international has long been linked with the Old Trafford outfit, and Red Devils boss Jose Mourinho is keen to add the player to his squad.
And now Griezmann has revealed there is a real chance of him moving to the Europa League finalists, with his future set to be finalised in the next fortnight.
"My future will be decided within two weeks, " he told French television show, Quotidien.
He was then asked what the possibility of him joining United this summer would be out of 10, he replied "six".
Some reports have suggested Griezmann will only sign for United if they can secure Champions League football for next season.
But Goal understands this is not the case, meaning defeat to Ajax on Wednesday evening would not necessarily end their chances of bringing Griezmann to the Theatre of Dreams.
The former Real Sociedad forward would likely cost in the region of €80 million should United make a bid.