The deal being discussed included outfielder Alfonso Soriano, the source said; however, sources told ESPN.com's Jayson Stark on Tuesday that the Dodgers were on the verge of acquiring Shane Victorino from the Phillies. That would likely take them out of the running for Soriano.
Cubs manager Dale Sveum said before Monday's game against the Pittsburgh Pirates that he would be surprised if Dempster is still a Cub after the deadline.
"Yeah I'd be surprised, I guess," Sveum said. "Yeah I would.
"There's just too many teams, and he's too good a pitcher for people not to want him these last two months."
Soriano told ESPNChicago.com on Sunday that he had not yet been approached by the Cubs about waiving his 10 and 5 rights, which gives him veto power over any trade as a veteran of 10 major league seasons and the past five with the same team. In the sixth year of an eight-year, $136 million contract, Soriano is still owed $44 million. How much of that $44 million the Cubs are willing to eat could determine whether he gets moved.
With the non-waiver trade deadline approaching at 3 p.m. CT Tuesday, Soriano, 36, said he would consider a trade depending on the team and the city involved.
"It will have to be a good city with a good team," Soriano said Sunday. "I would give the team the opportunity to trade me, but as I said the team and city would have to be good. Nobody has come to me about a trade yet, so we have two or three days left. We will just wait and see what happens."
The San Francisco Giants have also made inquiries about Soriano, according to another major league source.
The Dodgers, who have also shown interest in pitcher Matt Garza, have been high on Dempster's list since he was asked to give the Cubs the teams he would consider going to in a June meeting with president of baseball operations Theo Epstein.
Dodgers general manager Ned Colletti said he was optimistic the club could "do at least one thing" before Tuesday's non-waiver trade deadline.
"We've got a handful of clubs that we've been in a lot of communication with the last couple of weeks," Colletti said Monday evening before the Dodgers game against the Diamondbacks. "We're coming down to the finals days and final hours of it so we'll find out who's real now."
Colletti said the Dodgers were still in the market for starting pitching, bullpen help and another bat.
When asked how long the Dodgers could wait before making a final decision on some of the trades they are currently considering, Colletti seemed to indicate it was more a question of how long other teams were willing to wait.
"You could wait until tommorrow morning, probably. It depends how strong their ability to wait is," said Colletti, who famously traded for Manny Ramirez one minute before the deadline in 2008. "How much action they've got. How many choices they have. If they've got no other choices then ..."
The Cubs had a deal in place on July 23 with the Atlanta Braves for Dempster, but he did not consent to the trade initially, causing the Braves to take the deal off of the table the next day.
Dempster is scheduled to start for the Cubs on Tuesday against the Pittsburgh Pirates if he's not traded by then.
"Well, I'm preparing for it, and we'll see if anything changes from there," Dempster said Monday. "Baseball is a game of adjustments, and if that's not the case then I'll have to adjust and go from there. But for right now that's my goal, and that's what I'm trying to get ready for."
Asked if he has a gut feeling about where he will end up, Dempster said, "I have all kinds of gut feelings right now. I'm just trying to focus and narrow my focus to doing what I need to do to do my job if that's what is going to happen (Tuesday)."
Information from ESPNChicago.com's Doug Padilla and ESPNLosAngeles.com's Ramona Shelburne contributed to this story.