But he says that he has no idea if they will be able to get the deal over the line.
Mourinho says that United chief Ed Woodward is in charge of transfer dealings over the coming months – and he has been given a shortlist of players to secure.
Griezmann has previously revealed that there is a six out of 10 chance that he leaves Atletico Madrid for Old Trafford this summer.
United’s return to the Champions League next season, courtesy of their Europa League success, is added incentive for the French international to complete the move.
Mourinho is attending the Monaco F1 Grand Prix this weekend as he begins his summer holidays.
But when asked about the possibility of Griezmann joining United by Sky Sports, he said: “I have no idea. You have to ask Ed Woodward.
“He knows since March who I want, what I want and how much I want.
“He knows everything since March and now it is for him to work, because he will have holidays in August and in August I am working.”
A number of reports have suggested that Griezmann has agreed personal terms with the deal set to be rubber-stamped.
However, the he took to Twitter on Friday evening to reject those reports while admitting his future is uncertain.
“All the rumours are unfounded,” he wrote. “I’m still Colchonero. My direction will be established after discussion with my sporting advisor.”
Mourinho is now officially off-duty and he says that his focus is now off football.
Speaking during Monaco qualifying, he said: “Today is the day of the FA Cup final, the Spanish cup final, Portuguese cup final, World Cup Under-20s – and I don’t care. I don’t care at all.
“I am not even interested in results.
“I am really happy with our season, I am really happy with the trophies, I am really happy to be back to Champions League.
“But now is time to have a rest, disconnect completely, do other things and enjoy my family, my friends, enjoy some other sports that I love and forget football. I need that.”