United and Cavani have the option to extend his contract by another year, but both need to agree to activate it.
A source close to the player told Dawson that Cavani is now open to staying after initially feeling unhappy in Manchester. His father, Luis, told Argentinian media in March that the 34-year-old was upset with the suspension he received for using a racial term on social media.
Boca Juniors also reportedly expressed interest in signing Cavani, offering him the chance to be closer to his family in Uruguay.
But his tune appears to have changed. Now fully recovered from a nagging muscle injury, Cavani has become a more prominent fixture in Ole Gunnar Solskjaer's starting lineup. He's repaid the club with five goals in his last five matches, including a brace in Thursday's 6-2 thrashing of Roma in the Europa League.
Solskjaer said he's already held "open" and "honest" meetings with Cavani.
"I promised him Old Trafford and Manchester is a different place with fans in the stadium," the Norwegian told reporters.
Solskjaer added: "I'm doing my best. ... Hopefully he can see himself being here for another year."
Cavani joined the Red Devils from PSG last October on a free transfer.