Barcelona appeared close to landing the France World Cup star - but their failure to make an official move has opened the door to United.
Griezmann, 28, gave the Old Trafford outfit hope when he claimed that his future may now be away from Spain.
Speaking after France's 4-0 Euro 2020 qualifying win in Andorra on Tuesday, he said: "I don't know if I will stay in La Liga.
"I know where I want to go. I am also keen to get this over with as quickly as possible."