Kerry was at 49 percent, Bush at 42 percent and Nader at 2 percent in an American Research Group poll. Seven percent of the voters in the poll said they were undecided.
In a two-way matchup, Kerry led Bush by 50 percent to 43 percent.
The Ohio results come as Bush is registering some of the lowest ratings of his presidency in national polls and as the Iraq prisoner abuse scandal has the White House on the defensive.
The poll found that slightly more voters have a favorable view of Kerry, 44 percent, than unfavorable, 36 percent. A majority, 52 percent, had an unfavorable view of Bush. Almost two-thirds, 63 percent, had an unfavorable view of Nader.
Ohio is one of several states in the region that have lost manufacturing jobs under Bush, while Kerry has made the jobs issue central to his campaign.
Bush won Ohio in 2000 by 4.4 percentage points.
The poll of 600 likely voters was taken May 10-12 and has a margin of sampling error of plus or minus 4 percentage points.