In the poll released Thursday by CNN/ORC, just over half the respondents, 52 percent, said they support the president, while 45 percent would back Romney.
"Among independent voters, the poll indicates President Obama has a 53 percent-42 percent lead," CNN Polling Director Keating Holland said. "The president holds a nine point advantage among women voters and a smaller six point edge among men."
The poll found Republicans are somewhat more likely now to say they strongly support Romney than they were a few months ago. But his unfavorable rating has jumped to 48 percent among all voters, with only 47 percent viewing him favorably.
More than half, 52 percent of independents, now have an unfavorable view of Romney, up from 40 percent in May. That may be because 68 percent of independents and 64 percent of all voters think Romney supports policies that will help the rich rather than the middle class.
One question is whether Romney's choice of a vice presidential candidate can help the arithmetic. The CNN poll found that most of the people named as possible choices are still comparatively unknown.
Sen. Marco Rubio of Florida had the most name recognition and is the only person Romney has acknowledged is on the list. But almost half of voters said they know little about him -- and that includes 40 percent of Republicans.
ORC International conducted a telephone survey of 1,010 adults on Tuesday and Wednesday, including 911 registered voters. The margin of error is 3 percentage points.