It is anticipated that Girardi's boosted salary would sit within the $4- $5 million window, permitting him to become the second-highest paid manager in the Majors. The Angels' Mike Scioscia signed a 10-year, $50 million contract in 2009.
"The most important thing is he knows we'd like to have him stay and continue as manager of the New York Yankees as we move forward," Cashman said this week.
Cashman and Girardi met over coffee on Monday in New York, and Cashman also had meetings with Girardi's agent, Steve Mandell, this week.
Girardi said during his end-of-season news conference in Houston that he has enjoyed his time in the Yankees dugout, compiling a 564-408 record during the regular season while piloting the club to its 27th championship in the 2009 World Series.
Girardi is under contract through Oct. 31, and the Yankees are unlikely to grant other clubs permission to speak with him. The Cubs are thought to be interested in Girardi's services, while the Nationals and Reds also could join that group.
Yet, Girardi has said that his family now has roots in the New York area. The Peoria, Ill., product downplayed the suggestion that the Cubs represent a dream job, noting that his family ties to Chicago are not as strong as they once were.
Girardi later added that he expected a quick resolution to his contract status.
"It's not in my personality to drag things out," Girardi said. "I've always been a guy that likes to know what I'm going to do the next day."
Bryan Hoch is a reporter for MLB.com. Follow him on Twitter @bryanhoch and read his MLBlog, Bombers Beat. This story was not subject to the approval of Major League Baseball or its clubs.