Dempster, one of the top arms available as the July 31 non-waiver Trade Deadline nears, said he's had a few conversations with Cubs brass about potential moves, but no deal is imminent.
As a player with 10 years of Major League service and the last five with the same team, Dempster can veto any trade.
"It's just kind of like, 'Hey, there's teams interested,' and that's great. Nothing imminent, not that I know of," Dempster said. "I'm sure if they have something, they'll come to me. But right now I'm just trying to get ready for tomorrow's game."
Dempster's stock is at an all-time high, as the right-hander -- who is 5-3 with a 1.86 ERA on the season -- has tossed 33 consecutive scoreless innings. Dempster said he's conscious of the streak, and said he "[wants] to shatter" Orel Hershiser's all-time mark of 59 consecutive scoreless innings, set in 1988.
"I know I haven't given up a run in five starts. I'm not naive," Dempster said. "Like I said when I had a no-hitter going before, I know when I have a no-hitter going. I know that I have it, so I'm just going to try to keep it going, for sure. When you don't give up any runs, you win games, so I like that."
Between the scoreless-inning streak and preparing for his next start, Dempster said he doesn't spend much time keeping up with the rumor mill. Although he said it's flattering so many teams are reportedly interested in him, he's still focused on his next start, which -- as of Thursday afternoon -- appears will take place in a Cubs uniform.
"I just think that there's a lot of things between pitching and on-the-field stuff and then stuff about trades, and it's a lot of information to try to process, so for me, I try to limit what I read and my focus is just going out there and pitching," Dempster said. "Probably my biggest focus right now is just going out there and be ready every fifth day. I still have a job to do, and I take a lot of pride in that, so if I'm not putting my attention, my sole focus into that, I'm doing the team a little bit of a disservice."