The Spaniard is out-of-contract in July and is yet to agree a deal to stay at the club, with concerns growing that the likes of Barcelona could lure Mata away this summer.
Speculation that he could be offered fresh terms has heightened following Uniteds downturn in form, and journalist Andy Mitten, writing for SCMP, claims that Woodward invited the former Chelsea man into his office after Sundays defeat to hold talks about extending Matas stay with the Red Devils.
Manager Ole Gunnar Solskjaer reportedly wants to keep the 30-year-old, but Mata, who has often been on the periphery of the team, wants assurances over his place in the side and has already seen close friend Ander Herrera opt to join Paris Saint Germain on a lucrative contract.
Mata was an unused substitute in Uniteds shambolic 2-0 defeat to Cardiff on the final day of the season as Solskjaer opted to give some of the clubs youngsters an opportunity to impress.
The stumbling block could be Uniteds reluctance to offer players in their thirties anything more than a one-year deal with an option for another 12 months. Barcelonas reported interest in a two-year deal could force the United chiefs into a change of policy.
United said goodbye to Matas close friends Herrera and Antonio Valencia on Sunday with more players expected to follow them out of the exit door this summer.
The likes of Alexis Sanchez, Marcos Rojo, Paul Pogba, Romelu Lukaku and Matteo Darmian could all be part of a summer mass exodus as Solskjaer looks to change the recent strained recruitment strategy and have more of an impetus on young promising talent.
Mata has always been a fan favourite at Old Trafford due to his articulate straight talking weekly Monday blogs, not forgetting his breath-taking scissor-kick in Uniteds 2-1 victory at Anfield in 2015.