The new poll released Friday found Abbott with 48 percent support of those surveyed while Davis got 40 percent. Another 3 percent said they backed another candidate and the last 9 percent said they were undecided.
Rasmussen's new findings contrast with a previous survey it did in March which found Abbott leading Davis by double digits, 53 percent to 41 percent.
The race is still widely considered to favor Republicans.
The survey was conducted among 850 likely voters in Texas between August 4 and August 5. It had a margin of error of plus or minus 3.5 percentage points.