The poll, released Tuesday, found Davis, the state House minority leader, leading Brownback 51 percent to 41 percent. Another 3 percent said they preferred another candidate and 5 percent said they were undecided. The poll, by the Republican-leaning Rasmussen, is the latest in a series of signs showing Brownback in serious trouble in his re-election campaign.
Brownback's little-known primary challenger, Jennifer Winn, did surprisingly well in the primary against the governor, despite her long odds.
Before this Rasmussen poll, the TPM Polltracker average gave Davis a 6 point lead over Brownback.
The poll was conducted among 750 likely voters between August 6 and August 7. It had a margin of error of plus or minus 4 percentage points.