The Sunday Express says Conte is ready to go above the £86million release clause in Griezmann's contract to head off interest from Manchester United.
And that means splashing out more than the £89m that United paid Juventus to take Griezmann's close pal Paul Pogba back to Old Trafford.
Atletico were last week handed a transfer embargo from FIFA for the next two transfer windows.
Chelsea's Diego Costa, a long-term target for Atletico, could be used to sweeten the deal but Atletico would need to successfully appeal to the Court of Arbitration for Sport for that to even be an option.
United's interest in Griezmann was unearthed almost as soon as the transfer window ended, with manager Jose Mourinho keen to get working on plans for next summer.