BURKE, Va. — President Obama and challenger Mitt Romney, in dueling rallies on opposite ends of this tossup state, appealed for support Thursday from a large voting bloc that could prove crucial: military members, veterans, and their families.

While Obama is leading in most polls in Virginia, Romney’s push for a more muscular foreign policy and increases in defense spending has helped him maintain a traditional GOP advantage among members of the armed forces and veterans. The Republican nominee pressed that strength at an American Legion Post in Springfield, in Northern Virginia, accusing Obama of shirking his responsibilities.

“How in the world, as commander in chief, you could stand by as we shrink our military commitment financially is something that I don’t understand, and I will reverse it,” Romney pledged.

The Obama campaign has been working hard to chip away at Romney’s support by highlighting the president’s achievements on national security, including ending the unpopular war in Iraq, improving relations with allies, and killing Osama bin Laden.

He also has been promoting efforts to improve the care of returning service members, including reducing the wait time to see specialists and hiring more mental health specialists.

‘Our volunteer base is phenomenal in this area. We have had overwhelming support from the veterans community.’

In a rally in Virginia Beach, Obama was introduced by Senator Jim Webb, a veteran and former Navy secretary who wrote the new GI Bill to provide tuition benefits to returning veterans.

“Let’s make sure we are implementing the post-9/11 GI Bill,” Obama said to bursts of applause.

The Obama team has been making a concerted effort to reach out to military communities in the Old Dominion and other states, a grassroots effort that appears to be changing at least some minds.

“This November I am definitely going to support Obama,” said Tyre Nelson, 42, a 20-year Navy veteran from conservative Chesapeake, Va., who has previously voted Republican.

Nelson rifled through issues he deems paramount: “Veterans issues, wounded warrior issues, retiring service member issues, family issues.’’

Nelson expressed particular concern about the potential implications of the austere budget plan of Romney’s running mate, Representative Paul D. Ryan.

It “means an $11 billion cut to [veterans affairs] if they don’t take money from somewhere else,” Nelson said.

The Republicans, however, insist they are not giving up ground easily.

“We are definitely working hard to get the vote out with veterans,” said Ken Longo, 50, a 24-year Coast Guard veteran and chairman of the Republican Party in Virginia Beach, where he estimates a full 22 percent of the voters have served in the armed forces.

Every Wednesday evening, he said, the local GOP organizes a phone bank where veterans call fellow vets and their families and try to persuade them to vote for Romney and other Republicans on the ballot.

“Our volunteer base is phenomenal in this area,” added Joshua Baca, the Romney campaign’s national coalition director. “We have had overwhelming support from the veterans community.”

Earlier this summer, a Gallup poll showed Romney was leading Obama 58 percent to 34 percent among veterans nationally.

In the 2008 presidential election, Republican candidate Senator John McCain outperformed Obama among veterans by a full 10 percentage points, though that was lower than George W. Bush’s 16-point advantage over John Kerry in 2004.

Virginia, with 13 electoral votes, is flush with military personnel and veterans clustered around bases in Norfolk and Virginia Beach in the southeastern part of the state as well as communities like Burke that dot the Washington suburbs.

Virginia boasts more than 800,000 veterans, or more than 10 percent of the population. While that is only the fourth highest number of veterans per capita, the top three — Alaska, Maine, and Montana — have small populations that, combined, account for only 10 electoral votes.

The military vote is also being sought by both sides in tossup states such as Ohio, where Gulf War veteran Douglas D. “Douger” Lee, 44, of Columbus, said he would be voting for Obama because of the dramatic improvements he has seen at the VA.

“I used to be practically spit on,” said Lee, who suffers from post-traumatic stress. Once, Lee said, he went to the VA in a suicidal state and was put alone in a room for 45 minutes. “It was terrible, they would treat you like you were an inmate in jail.”

The experience now, he said over a beer at a Columbus bar, is far different.

“The last time I was in they actually treated me like a human being,” he said.

“I don’t know what they did to fix it so fast,” he said, but he credits the Obama administration with making veterans a higher priority.

Among a series of investments made by the Obama administration was $1 billion added to the 2009 stimulus package to upgrade VA hospitals.

Also, the Pentagon and Department of Veterans Affairs have instituted a series of programs to help educate soldiers preparing to leave the service on how to secure health benefits and job training.

Yet many veterans are not experiencing the same efficiency and compassion that Lee found in Ohio.

An increase in requests for disability and education benefits has increased the backlog — and lag time — for veterans in many parts of the country, despite a hiring spree by the administration.

The GOP is relying heavily on high-level surrogates to spread Romney’s message here.

For example, Longo recently organized a talk by former mayor Rudolph Giuliani of New York at a VFW post in Virginia Beach, which drew a full house.

Other Romney surrogates who have participated in round tables and town halls in Virginia include McCain and Terrie Suit, Virginia’s secretary of veterans affairs and homeland security.

Republicans have blamed Obama for pending cuts in defense, a sector critical to Virginia’s economy, even though the Republican-led House of Representatives also backed the deficit reduction package last year in an effort to avoid defaulting on the national debt.

“It is still a troubled and dangerous world,’’ Romney said Thursday. “And the idea of cutting our military commitment by a trillion dollars over this decade is unthinkable and devastating.’’

Romney has proposed substantially increasing the production of warships from nine to 15 per year, some built in Virginia shipyards.

Another major issue highlighted by Romney supporters is the Pentagon-run health insurance program known as Tricare.

The Obama administration has proposed increasing premiums, which have not gone up in a decade, and basing them on the overall income of the insured.

Romney has insisted he would not increase such costs.

But some in the Romney camp complain that he may be missing an opportunity by not making greater efforts to reach what should be a natural voting bloc for him.

“We haven’t been asked to do very much,” said retired Army Colonel Ken Allard, a member of the national advisory board of the official Veterans and Military Families for Romney Coalition.

“If this guy doesn’t stand on hind legs and start punching people this thing is over.”

The Obama campaign has a robust network of organizers in each state.

Rob Diamond, a retired Navy officer who oversees the campaign’s veterans outreach, estimated the Obama team has signed 20,000 volunteers for “direct voter persuasion” — teams of vets canvassing neighborhoods in Virginia and other critical states.

But even some Obama’s supporters acknowledge it is a tough sell.

“It’s an uphill job here,” said Jack Carter, 72, who served as a Green Beret in Vietnam and now lives near Little Creek, where the Navy SEALs who killed bin Laden are based.

He said many younger troops seem open to voting for their commander-in-chief, but older soldiers and retirees often cling to a belief that the Republican candidate will be a better advocate.

“We tell them, ‘Look at the facts and see if you really want to stay with the other guy,’’’ Carter said. “We may have a chance to sway some to our side.”