United have been linked with at least two huge bids for the teenager so far this year.
And Marca says days before Fati was formally promoted to Barca's first team squad and his contract upgraded, a new offer arrived for the teen from an unnamed club.
The proposal was for €125m up front with a further €25m due in bonuses, but Barca's board turned it down.
It's been suggested in Spain that United were the club in question to make the bid.
Fati's new deal has now seen his buyout clause jump from €170m to €400m.