Rep. Paul Ryan’s selection as Mitt Romney’s running mate has the power to reshape the race for control of Congress, with the already-simmering issue of the Wisconsin Republican’s budget now firmly at the forefront of the 2012 election.

Democrats have gotten significant mileage out of attacking the budget Ryan has proposed as chairman of the House Budget Committee, particularly the portion that aims to turn Medicare into a voucher program.

Within minutes of Ryan’s selection Saturday morning, President Obama and Democrats on Capitol Hill seized on the issue.

“His plan . . . would end Medicare as we know it by turning it into a voucher system, shifting thousands of dollars in health-care costs to seniors,” Obama said in a statement on Ryan’s selection.

Rep. Steve Israel (D-N.Y.), chairman of House Democrats’ campaign committee, echoed Obama: “Congressman Paul Ryan and Mitt Romney are a match made in millionaires’ heaven, but they’ll be a nightmare for seniors who’ve earned their Medicare benefits.”

In recent years, Democrats have used Ryan’s budget to argue that Republicans will “end Medicare as we know it” — a line of attack that has proved potent in some races. Leaders of both parties acknowledge that it helped Democrats win recent special elections in Republican-leaning districts in New York and Arizona.

Democrats say putting Ryan on the ticket will make his budget even more of a focal point in 2012 and force Republicans who haven’t already taken a position on it to do so.

“Our country needs to move forward, not rehash failed ideas,” said Rep. Kathy Hochul (D-N.Y.), who won the upstate New York special election last year and is already attaching her challenger’s name to Ryan’s Medicare proposal.

Poll after poll shows that Americans, while wanting to cut waste in the budget, are strongly resistant to any major cuts or changes to federal entitlement programs such as Medicare.

Nonetheless, Republicans believe they have been able to neutralize the issue by pointing to hundreds of billions in Medicare cuts in President Obama’s health-care law. And even though Ryan’s budget has put many Republicans in an uncomfortable spot politically, they believe the effects of putting him on the ticket are limited because they were already going to be attacked on the issue.

“On a candidate level, it doesn’t change anything that wasn’t already the case yesterday,” said Brian Walsh, a spokesman for the Senate GOP’s campaign committee. “Again, every Democrat already voted to cut Medicare; they don’t have a leg to stand on here.”

Republicans argue that their strategy has already paid dividends, pointing to a sizeable victory in another recent special election for a Nevada House district that split evenly between Obama and Sen. John McCain (R-Ariz.) in the 2008 presidential race.

“We wanted to fight Medicare to a draw but actually won the issue by Election Day,” said Rob Stutzman, a GOP consultant who worked on the Nevada race.

There is some evidence to support the Republican optimism on the issue.

A recent poll from Democratic pollster Democracy Corps showed that when the Ryan budget is described as attempting to “save Medicare” in battleground House districts — many of them Democratic-leaning — 52 percent of voters say they support it, while 37 percent oppose it.

Republicans including Romney have quickly moved to incorporate that argument into their campaigns.

“Unlike the current president, who has cut Medicare funding by $700 billion, we will preserve and protect Medicare and Social Security,” Romney said in announcing Ryan as his pick Saturday in Norfolk, Va.

The same poll, though, showed that when it is argued that the Ryan budget “slashes and privatizes Medicare,” 66 percent of voters say they have “serious doubts” about it.

Even when the language is more neutral, Americans are resistant to the idea of Medicare being transformed into a voucher program. A Pew Research Center poll last year showed 34 percent favored a voucher program, while 65 percent preferred that the program remain as-is.

Democrats need to win 25 House seats to retake the chamber, and they need to lose no more than three Senate seats to keep their majority there. The Senate is considered to be more in play in 2012, with Democrats defending many conservative-leaning and swing states.

Political analysts generally agree that the Ryan budget poses problems for the GOP. Republicans have had to vote for it and press the issue, they argue, because the conservative base expects them to do something major on the budget with their newfound House majority, not necessarily because it’s good politics.

“From a political standpoint, Medicare reform is very dangerous territory. House Republicans are not just pushing the envelope — they are soaking it with lighter fluid and waving a match at it,” nonpartisan analyst Charlie Cook wrote when the GOP first voted on Ryan’s budget.

Felicia Sonmez contributed to this report.