The Red Devils have been heavily linked with the France international for several months, with interest first appearing during the winter window.
A big-money move has been mooted for the summer, with Griezmann having recently opened the door to a possible deal.
He told French TV programme Quotidien that his chances of joining United are “six out of 10”, before then stating in L’Equipe that he is “ready to go” and challenge for major honours.
Mourinho, though, told BeIN Sports after overseeing a Europa League final victory on Wednesday: "I do not want to talk about Griezmann because that would be disrespectful to his club, Atletico.
"Just how I do not like others talking about by my players, I'm not going to talk about Antoine either."
Mourinho did, however, admit to the BT Sport that he has already identified targets.
United are expected to be busy in the summer window, as they seek to build on their cup successes this season by challenging for the Premier League title in 2017-18.
Executive vice-chairman Ed Woodward is the man charged with the task of delivering the goods to Mourinho, with the identity of those he has been challenged to get set to become apparent in the coming weeks.
"Ed Woodward has my list, what I want, what I would like for more than two months," Mourinho said after guiding United to a 2-0 victory over Ajax.
"So now it is up to him and the owners."
United also have decisions to make on the likes of Zlatan Ibrahimovic and Michael Carrick and those approaching the end of their current contracts.