AS says United have made a 'dream' proposal to the Brazil international, who has been unsettled by the approach.
Casemiro penned new terms with Real Madrid 12 months ago to 2025, with a buyout clause set at €1bn.
But United's offer has turned the player's head and is raising concern inside Real Madrid.
Casemiro, who has just turned 30, is torn between his commitment to Real or securing a mega bucks contract which would run to his retirement.
United, meanwhile, plan to offer €60m to Real to convince them to sell.