The Daily Mail says Griezmann, 26, will be paid £235,000 ($297,000) a week as part of his new contract, less than the £300,000 United were reportedly willing to offer him.
However, the France international's buyout clause has been kept unchanged at €100m.
United were keen on signing Griezmann but their interest cooled after a transfer ban on Atletico was upheld by the Court of Arbitration of Sport on 1 June.
The ruling meant the Spanish club would not be able to replace the 26-year-old if he moved to Old Trafford.
Griezmann is now Atletico's highest-paid player and the fourth-highest paid player in La Liga after Cristiano Ronaldo, Lionel Messi and Neymar.
The France international said last month that he was ready to leave Atletico in order to win titles. The Madrid club finished third in La Liga last season and exited the Champions League at the semi-final stage.
"Today, if I have to move it will be no problem," he told French daily L'Equipe. "I'm ready to go.
"I want to win titles. We finished third in La Liga, it was the objective of the club, but we, the players, want more.
"Winning titles is what I will look for this summer when deciding on my future."
But after Atletico's transfer ban was upheld, Griezmann posted a message on social media saying: "Now more than ever! #atleti #alltogether."
United have moved on from Griezmann to other targets, with the club expected to make a big push to sign Alvaro Morata from Real Madrid this week.
Benfica centre-back Victor Lindelof, 22, became Jose Mourinho's first signing of the summer transfer window over the weekend.
The Red Devils have also been linked with Everton striker Romelu Lukaku and Torino forward Andrea Belotti.