Manchester United want to sign Griezmann by triggering his £85m buy-out clause this summer.
Chelsea and Paris Saint-Germain are also keen on the France international.
But Atletico Madrid want to keep their best player as they prepare to move into their new stadium next season.
Antoine Griezmann's next club Wed, April 5, 2017
Which club is favourite to sign Atletico Madrid ace Antoine Griezmann?
Play slideshow Getty Images 1 of 11
Which clubs are tipped to sign Atletico Madrid ace Antoine Griezmann?
And, according to the Daily Mirror, Atletico will insert a new £130m release clause into his new contract.
It is claimed Griezmann is aware of how much he can earn at Old Trafford, with Paul Pogba signing a £290,000-a-week deal last summer.
Griezmann currently earns £120,000 a week but the report adds Atletico are willing to offer as much as £180,000 a week to keep him happy.
The deal would also tie Griezmann down until 2023, by which time he would be 32 years old.
Manchester United want to sign striker Antoine Griezmann
Atletico Madrid want Griezmann (left) to sign a new £180,000-a-week contract
Get Quotes on Home Insurance
Griezmann is a regular in the France national side and had a stellar Euro 2016 campaign
Asked after the 1-1 draw with Real Madrid if Griezmann would remain at Atletico, manager Diego Simeone said: "I don't know where he's going to go, but I'm thrilled with how he plays, with his enthusiasm.
"It's a pleasure to watch him turn in performances like he does.
"I was discussing his position with a lot of people. This lad is sharp, he heads well, he turns well with the ball.
“What's happening to him right now makes me happy."