The survey of 1,002 registered voters showed 47 percent would vote for McDonnell, a former state attorney general, while 40 percent would back Deeds, a state senator. One percent said they'd support someone else, 3 percent would not vote and 9 percent offered no opinion.
The poll was conducted through telephone interviews from Aug. 11-14. Its primary result is outside the poll's margin of error of plus or minus 3 percentage points.
Among voters who say they are certain to vote, 54 percent say they support McDonnell while 39 percent back Deeds. Seven percent offered no opinion.
Virginia and New Jersey are the only states to elect governors in this off-year election -- the first statewide elections since President Barack Obama and congressional Democrats consolidated their hold on national power last fall.
The poll, however, found that Obama's endorsement has been a wash for Deeds. Thirty-four percent said they were more likely to back Deeds because of the president's endorsement, the same share said it made them less likely to support him, and 30 percent said it made no difference. Three percent offered no opinion.
It also found that Virginians have little interest in this year's election. Only 49 percent said they were following the Deeds-McDonnell race closely compared to 60 percent who said they were closely following the governor's race between Republican Jerry W. Kilgore and Kaine at a comparable point in the 2005 election.
Four years ago, Kaine benefited from the perception by 65 percent of registered voters the Post surveyed that the state, headed then by Democratic Gov. Mark R. Warner, was on the right track. This year, respondents were evenly split: 47 percent said Virginia is on the right track, 45 percent don't like the state's direction, 7 percent had no opinion.
Fifty-six percent of those surveyed approved of the job Kaine is doing less then five months before he leaves office compared to 36 percent who disapproved. Eight percent had no opinion.
Warner, now in his freshman year in the U.S. Senate, had a 67 percent job approval mark while Democratic Sen. Jim Webb got a 51 percent job approval rating.