President Barack Obama and Democrats are counting on regaining support from older voters who switched to the GOP in 2008 and 2010 by attacking Republican plans to revamp Medicare. But Mitt Romney is proving to be a formidable competitor in this battle.

The Republican presidential front-runner has drawn large shares of older voters during the primaries, and recent polls show him ahead of Mr. Obama among seniors in swing states such as Ohio, Pennsylvania and Florida.

The battle for seniors is being fought in large part over the House Republican budget authored by Rep. Paul Ryan (R., Wis.). It calls for seniors, starting in 2023, to choose among private health-insurance plans or the traditional government-run option and receive aid from the government for their insurance premiums. The new model would apply to people who currently are under age 55.

Democrats are doing what they can to link Mr. Romney to the Ryan Medicare plan. Mr. Obama noted in a speech last week that Mr. Romney called the Ryan budget "marvelous," and said the proposed Medicare changes in particular are "a bad idea, and it will ultimately end Medicare as we know it."

Republicans say the Ryan Medicare plan would shore up the program's shaky finances and preserve it for future generations, without affecting current retirees' benefits. They also believe they can inoculate themselves against charges they are cutting Medicare, because Mr. Obama's 2010 health-care law included provisions that curbed Medicare spending growth by some $500 billion over 10 years.

Republicans have noted that Mr. Obama's limits on Medicare growth helped the GOP win a special House election in Nevada last year.

Mr. Romney took that line of attack as he campaigned before last week's Wisconsin primary with Mr. Ryan at his side. "You'll see signs during the campaign: 'Republicans keep your hands off my Medicare,'" he said at a town hall in Middleton, Wis. "Hey guys, go back and meet your own president. It is the president who went after Medicare."

Mr. Obama and fellow Democrats are trying to reverse a growing trend: Over the last decade, voters over age 65 have increasingly turned to the GOP, in stark contrast to the Democratic-leaning "Greatest Generation" that preceded them. In 2008, Mr. Obama improved on his party's 2004 showing among every age group—except among seniors.

An analysis by the nonpartisan Pew Research Center for the People and the Press concluded that the over-65 set is now more conservative on social issues, angrier about the direction of the country and more uneasy about the growth of diversity in the U.S. than younger generations.

Seniors favored GOP nominee John McCain in 2008 by 53% to 46% for Mr. Obama. Things got worse for Democrats in 2010, when older voters favored GOP candidates by 59% to 38%.

Seniors have been drawn to support Mr. Romney in GOP primaries in part because he has focused so much on the economy—the top concern of older voters, surveys find. Older people also tend to be more change-averse, and many are reassured by Mr. Romney's reputation for pragmatic competence. "He did a good job in Massachusetts and I know he did an excellent job with the Olympics," said Alice Allen, a retired Mississippi insurance agent who voted for Mr. Romney in her state's primary.

Last year, Mr. Ryan stirred controversy with a proposal to replace Medicare's fee-for-service system with a series of private insurance plans, which would receive federal subsidies and offer policies to seniors. The plan was rejected in the Senate.

Mr. Romney has embraced Mr. Ryan's latest budget. "A few common-sense reforms are going to ensure that we can make good on our promises to our seniors and we can also save Social Security and Medicare for the future generations," said Mr. Romney in a Detroit speech, when he also called for increasing the eligibility age for Medicare.

The Obama campaign has gone on the attack, sending Vice President Joe Biden to retiree-rich South Florida. "There's a fundamental difference between us and the Republicans: We believe in strengthening Medicare. They don't," said Mr. Biden.

Democrats are singing the same tune in congressional races. In the hotly contested Virginia Senate race, the Democratic Senatorial Campaign Committee is blasting former Sen. George Allen, a Republican, for speaking favorably about the Ryan budget a year ago.

When the House voted on the Ryan budget on March 29, Rep. Dennis Rehberg of Montana was one of the 10 Republicans who voted against it. He is running for the Senate and cited concerns about the Medicare provisions.

Some older voters are growing weary of the scare tactics. "I'm tired of the fear mongering," said Dawn Heilman, a retired teacher in Ohio who backs Mr. Romney. "If you are realistic you have to accept some changes."