While many teams are expected to emerge as interested buyers as the Deadline nears, the Dodgers, widely known to be Dempster's No. 1 choice, seem to be the frontrunners.
Whether Dempster stays or goes, Cubs manager Dale Sveum is ready for the Trade Deadline to be over.
"I think there's no question it's going to weigh on you a little bit," Sveum said of the distractions created by the rumors. "Everybody's human."
If the Dodgers can land Dempster, it would be the cherry on top of a great week. Having not only acquired Ramirez from the Marlins on Wednesday, Los Angeles is coming off a weekend sweep of the Giants that forged a virtual tie atop the National League West.
Shutting out the Giants in successive games for the first time since 1957, the Dodgers headed home for a three-game set against the D-backs beginning on Monday.
"Any time you sweep the Giants at home, regardless of what happened the previous series, it's a big deal," Sunday's winning pitcher, Clayton Kershaw, said. "We've got some momentum going back and we've got to be ready for the Diamondbacks."
While Dempster is heavily on the Dodgers' radar, so is Phillies center fielder Shane Victorino. With the Phillies getting swept over the weekend by the Braves to fall 12 1/2 games back of the second Wild Card spot, they are expected to be an active seller before the Deadline.
Meanwhile, the other L.A. team, the Angels, headed to Arlington for a Texas-size showdown with the Rangers. Sitting five games behind the Rangers in the American League West, the Angels will play four games against the division leader.
"It's going to be a fun series," Rangers outfielder David Murphy said. "It's always fun when we play those guys. We are in our home park and have home-field advantage. We definitely need to use that to our advantage."
Facing a stacked starting rotation, the Rangers will look to match the firepower of Ervin Santana, Jered Weaver, Dan Haren and C.J. Wilson in the series.
After fireballer Neftali Feliz was scratched from his rehab start with Triple-A Round Rock on Sunday because of elbow discomfort, the Rangers may target a high-profile starter on the market, notably Josh Johnson.
The Rangers, along with five other teams, are reportedly interested in Johnson. They were also one of 15 teams that scouted Johnson's start in Miami on Sunday against the Padres, in which the right-hander allowed one earned run on two hits in five innings.
In the AL Central, White Sox left-hander Francisco Liriano, acquired last week from the Twins, will face his former team right away. Liriano is to get the ball on Tuesday in the second game of a series between the clubs in Minnesota.
"It was a big surprise," Liriano said of being traded within the division. "I knew something like that was going to happen but not to the White Sox. It is what it is. They are a good team and in first place and they believe in me. I'm excited to help them win some ballgames."
Staying in the Midwest, the Reds will try to extend their 10-game winning streak on Monday, when they begin a four-game set against the Padres at Great American Ball Park in Cincinnati.
Winning six straight games on the road, including a 7-2 victory over the Rockies on Sunday, the Reds have padded their NL Central lead over the Pirates to three games.
"I'm always talking about the perfect road trip; finally had one," manager Dusty Baker said. "It's been a long time. I can't remember the last perfect road trip. The guys responded well."