As Mitt Romney heads into next week’s Republican National Convention, he narrowly trails President Obama in national polls but has closed the gap or taken a slight lead in several of the dozen swing states that could decide November’s presidential election.

According to the latest Associated Press-GfK poll, the race remains essentially a dead heat with slightly more than 10 weeks to go before Election Day, and Romney’s choice of Rep. Paul Ryan (R-Wis.) as his running mate appears to have made little difference. The poll, conducted Aug. 16-20, shows registered voters favoring Obama and Vice President Biden over the Romney-Ryan ticket by 47 percent to 46 percent — little changed from June when Obama led Romney by 47 percent to 44 percent.

A nationwide NBC News/Wall Street Journal poll conducted during the same time frame shows Obama-Biden leading Romney-Ryan by 48 percent to 44 percent. A majority of voters (54 percent) said Romney’s choice of the conservative Wisconsin congressman does not affect their vote either way, while 23 percent said it make them less likely to vote for the GOP ticket and 22 percent said it make them more likely.

A new USC Annenberg/Los Angeles Times poll, conducted Aug. 13-19 and released Thursday, has Obama leading Romney 48 percent to 45 percent among all registered voters surveyed, and 48 percent to 46 percent among those likely to vote, the newspaper reported.

The polls largely preceded the furor that erupted this week over antiabortion comments by Rep. Todd Akin (Mo.), a Republican running for a Democratic-held Senate seat. Akin ignited the controversy when he told a St. Louis television station Sunday that victims of “legitimate rape” rarely get pregnant because “the female body has ways to try to shut that whole thing down.”

A state poll conducted Wednesday by Rasmussen Reports showed that support for Akin plunged after his remarks, with Sen. Claire McCaskill (D) now leading him by 10 points, 48 percent to 38 percent. Before the uproar, Akin had led the Democratic incumbent by an average of five points.

Romney and other GOP leaders urged Akin to drop out of the race for the good of the party, and Ryan called him personally to make that appeal. The uproar drew attention to Ryan’s socially conservative views; he has shared Akin’s opposition to abortion exceptions for rape and incest, but he muted those beliefs this week in deference to Romney’s support for such exceptions.

The issue threatened to sidetrack what Romney hoped would be a Republican message focused intensely on the economy at the four-day convention beginning Monday in Tampa. “It’s a huge distraction,” Saul Anuzis, a Republican National Committee member from Michigan and a top Romney backer, told the Associated Press. “We should be talking about the economy, and here we are consumed by these side issues.”

The convention also faced a potential threat from Tropical Storm Isaac, which is expected to gain hurricane strength over the Caribbean by Friday and could hit Florida as an estimated 70,000 convention delegates, journalists and protesters converge on Tampa.

Officials in Tampa, a city on the western coast of Florida with more than 335,000 inhabitants, said they were prepared to take emergency measures and evacuate people if necessary. “Public safety will always trump politics,” Tampa Mayor Bob Buckhorn (D) said, AP reported. “And so my job, and our job, if we move into that mode, is to make sure we get people out of harm’s way.”

According to the AP-GfK poll, while Obama holds a slight edge over Romney among registered voters, most Americans — 58 percent overall, including more than a quarter of Romney’s supporters — expect Obama to win reelection.

A positive sign for Romney, the poll showed, is that he leads Obama among independent voters, 41 percent to 30 percent.

The former Massachusetts governor also could take solace from several new polls in swing states, including three released Thursday by Quinnipiac University in conjunction with CBS News and the New York Times. They show that in Florida and Wisconsin, Romney has cut into Obama’s lead, while Ohio remains unchanged.

In Florida, Obama leads Romney 49 percent to 46 percent, compared to 51-45 percent on Aug. 1. In Wisconsin, Obama’s lead is two percentage points (49-47 percent), down from six points (51-45 percent) on Aug. 8. In Ohio, Obama maintains a 50-44 percent lead, the same as on Aug. 1, according to Quinnipiac.

In each of the three states, the poll showed, voters thought Obama would do a better job on Medicare than Romney, and they rejected by wide margins a voucher-type system proposed by Ryan as part of a budget plan that would overhaul Medicare. Romney has expressed support for Ryan’s proposed budget. Voters disapproved of a voucher system by a 62-28 margin in Florida, 64-27 percent in Ohio and 59-32 percent in Wisconsin, the Quinnipiac/CBS/NYT poll found.

Other polls, notably a Quinnipiac survey in Colorado and a Purple Strategies poll in Virginia, show Romney opening a lead for the first time in those states. In Ohio, a Purple Strategies poll last week showed Romney ahead, contradicting Quinnipiac. And In Pennsylvania, a Franklin and Marshall College poll showed Obama’s lead over Romney at five percentage points, down from 12 in early June.