Chicago fired Dale Sveum on Monday, and sources say that Cubs ownership views Girardi as someone who can help put a jolt into fan interest in the team.
The Cubs' attendance has declined in each of the past five seasons, at a time when the organization is trying to rebuild and probably is at least two or three years away from contending again. Hiring Girardi in anticipation of increased fan interest could be more financially efficient for the Cubs than pursuing big-money free-agent players.
The Cubs' baseball operations department has only begun its search for the next manager and isn't locked into Girardi -- or anyone else -- as its primary option.
But sources say that Cubs ownership wants Girardi, is making a strong internal argument to hire him and intends to make him a serious contract offer, if given the chance.
The negotiations between the Yankees and Girardi could proceed quickly, given New York's intention -- as stated by Cashman to reporters Tuesday -- to offer him a raise.
Cashman declined to say if he would grant Girardi permission to speak to the Cubs. Because Girardi is under contract until Oct. 31, a source told ESPNNewYork.com that the Yankees are hesitant to give him permission to speak with the Cubs.
The highest-paid manager in the sport is Mike Scioscia, at $5 million annually, with Jim Leyland next at $4 million. The Yankees presumably will offer Girardi something close to the top of the market, more than the $3 million he earned annually under the terms of his current contract.
If Girardi rejects New York's offer -- to pursue the Cubs' job, or to work in television -- then the Yankees may rapidly move ahead with the search for their next manager.
From the Yankees' point of view, allowing Girardi to talk to the Cubs prior to Oct. 31 would only give Girardi more leverage in negotiations. If talks were to break down, the Yankees might let him talk to the Cubs.
As the Cubs wait for Girardi's situation to play out, they continue to keep other options on the table. Sources told ESPN.com that they've checked in with the San Diego Padres about assistant general manager A.J. Hinch, who previously managed the Arizona Diamondbacks to an 89-123 record during the 2009-2010 seasons.
Hinch, a Stanford product, spent seven seasons in the big leagues as a catcher with Oakland, Kansas City, Detroit and Philadelphia. He's one potential fallback option if things fail to work out with Girardi in Chicago. Although the Cubs have yet to ask for permission to interview Hinch, that's expected to be a mere formality.
Industry sources also said the Cubs' list of potential managerial candidates does not include former big league catcher Brad Ausmus, who is currently working as a special assistant in San Diego. While Ausmus is regarded by many as a future managerial star, the Cubs appear to be looking for someone with more experience in the dugout.
Girardi said Sunday that he would like his situation to be cleared up shortly.
Girardi is an Illinois native who attended Northwestern and played for the Cubs. He managed the Marlins in 2006, when he was named the National League Manager of the Year, before taking over the Yankees in 2008.
Information from ESPN.com's Jerry Crasnick and ESPNNewYork.com's Andrew Marchand was used in this report.