Mata is due to be out of contract in June and is on course to leave United as a free agent after spending five-and-a-half years in Manchester.
The Spaniard, a popular figure among fans, has been linked with the likes of Barcelona and Atletico Madrid in recent months and has been eligible to negotiate a pre-contract agreement with overseas clubs since January, when he entered the last six months of his current deal.
Mata had previously been unable to come to an agreement with United over an extension - the length of the proposed deal is rumoured to have been the sticking point - but the Manchester Evening News has now reported that United have submitted one last offer in a bid to keep him.
The story claims that Mata has been offered a fresh one-year contract with the option of an additional 12 months, potentially tying him to the club until 2021. It is also said that Mata is 'calm' about the situation and that United are 'more hopeful' he will agree to the deal.
Speaking last month, Mata's father, who also doubles as his agent, confirmed that United 'have offered to renew', while also stating that his son is 'happy' at the club.
Whether that offer is the same as the one now being reported by the MEN is unclear.
Mata Sr. additionally confirmed at that time that there have been proposals 'from Champions League teams', while the only clubs he ruled out were Liverpool and Manchester City.
"There are two options that do not [appeal]: Liverpool and City. We have to respect the Manchester United fans, he is held very dear here," Mata Sr. said.
He has only started 15 Premier League games this season, although he did score his first goal in any competition since January in last weekend's draw against former club Chelsea.