The source said Wright had been offered at least a seven-year extension on top of the $16 million he is owed for 2013.
A source familiar with the matter did not believe the deal would be finalized before next week's winter meetings. A second source said it was "doubtful" a deal would be completed this week and characterized all suggestions otherwise as "inaccurate."
"It could, but it's certainly speculative that it will," a different source added.
Wright, 30, became the franchise's leader in hits, runs, walks and RBIs during the 2012 season.
"There are offers going back and forth," the source said.
Mets right-hander R.A. Dickey, who has been in regular contact with Wright, expressed confidence Tuesday that the third baseman ultimately would remain with the Mets long term.
Also, with Friday the deadline to non-tender arbitration players, the Mets plan to cut loose three players this week -- right-handers Mike Pelfrey and Manny Acosta and center fielder Andres Torres, according to a major league source.
Pelfrey, the team's first-round draft pick in 2005, is rehabbing from Tommy John surgery performed May 1. While he is optimistic he will be ready well before Opening Day, team officials are more uncertain. Because Pelfrey made $5.785 million this season, the Mets plan to non-tender him, then consider re-signing him at a discounted price.
Information from ESPN.com's Jayson Stark was used in this report.