De Gea's current deal runs out in 12 months and efforts to get the 28-year-old to agree an extension have failed, BBC Sport reported.
It is understood United's latest offer would make De Gea the highest-paid goalkeeper in the world.
He came within a minute of joining Real Madrid in 2015.
Whilst it is felt the former Atletico Madrid man wants to remain in Manchester, it is understood his advisors believe his status as one of the key figures in Ole Gunnar Solskjaer's squad, and United's player of the year in four seasons out of five, should make him one of the club's highest earners.
Out-of-favor forward Alexis Sanchez's £390,000-a-week contract is seen as the benchmark.
Senior United figures have previously indicated they would allow De Gea's contract to run down in the hope an agreement will be reached, even though it risks one of their most saleable assets leaving for nothing at the end of the season.