In his return Sunday afternoon, the right-hander will try to extend that run and remain perfect in road matchups against the New York Mets while working within a limited pitch count.
Trade rumors have swirled around Dempster (3-3, 2.11 ERA) for weeks, and the right-hander remained an attractive commodity with three consecutive scoreless outings covering 22 innings. That streak matches Carlos Zambrano's from 2004.
"The trade rumors have been there. That stuff doesn't bother me. I'll just try to do my job the best I can and let those things take care of themselves," Dempster told the team's website. "I'll focus on going out there and taking it one start at a time, one pitch at a time and just go from there."
Dempster, however, was also dealing with tightness in his right lat during that streak. It eventually became too difficult to manage, and the Cubs (32-52) placed him on the disabled list June 18.
It's a tough (injury) to gauge because it's just a matter of how I felt," he said. "I started to feel better, especially the last week or so."
In his first start back, the 35-year-old right-hander is expected to be limited to 70 to 80 pitches. That could make it difficult to earn a sixth win in as many road starts against the New York Mets during his first such matchup since May 27, 2002.
Dempster is also seeking to become the first Cubs starter since Jim Bullinger in 1995 to record at least 23 straight scoreless innings. Bullinger eventually stretched that run to 24 and Greg Maddux tossed 25 2-3 consecutive shutout innings July 23-Aug. 1, 1990.
During his most recent start against the Mets, Dempster allowed one run and scattered seven hits over seven innings of an 11-1 rout at Wrigley Field on May 24, 2011.
Jonathon Niese (7-3, 3.35) took the loss in that game and faces Dempster again Sunday.
The Mets' left-hander fared much better in his last matchup with the Cubs, a 17-1 win June 27 at Wrigley in which he yielded one win and eight hits over seven innings.
Niese enters this start with a 1.69 ERA over three consecutive wins, and he gave up a solo homer and two singles over eight innings of Tuesday's 11-1 victory over Philadelphia. He also singled in two runs during that game.
"What he's been doing is really going after hitters," manager Terry Collins said. "He's stepped up and has started showing what he can do."
The Mets (46-39) used home runs from Ike Davis and Jordany Valdespin for a 3-1 victory in Saturday's middle game of the series. Davis is batting .309 with seven homers and 28 RBIs over the past 24 games after hitting .158 through his first 56.
"I'll take those homers and RBIs," Collins said. "Especially with his defense, because he brings something else to the table that's hard to replace."
Davis is 2 for 3 in his career against Dempster and Darwin Barney is 4 for 6 lifetime versus Niese. Cubs rookie Anthony Rizzo, batting .386 with four homers and nine RBIs in his first 11 games, doubled off the Mets left-hander in last week's loss.