The U.S. representative from Wisconsin wrote that the accounts would give workers an opportunity "to build a significant nest egg for retirement that far exceeds what the current program can provide." Workers 55 and older would stay in the current system.
Romney has not embraced the proposal, and Ryan, chairman of the House Budget Committee, did not include it in either of the federal budgets passed by House Republicans the last two years. But now that Ryan is running for vice president, the Democrats hope to capitalize on the issue.
Bush's proposal for private accounts got a chilly reception from members of both major parties in Congress, though Ryan embraced it. Democrats used the issue against GOP congressional candidates in the 2006 election, when they regained control of the House and Senate.
"The very last thing we ought to be doing is putting at risk the retirement security of millions of America's seniors," said U.S. Rep. Debbie Wasserman Schultz of Florida, who heads the Democratic National Committee.
Until now, Social Security had been largely absent from the presidential campaign. President Obama has yet to lay out a detailed plan for addressing the issue, and his silence is drawing criticism from advocates who supported him in the past. Romney has been more forthcoming with proposals, but Social Security has not been a big part of his campaign, either.
Romney, in his book No Apology, said he liked the idea of personal accounts. But, he wrote, "Given the volatility of investment values that we have just experienced, I would prefer that individual accounts were added to Social Security, not diverted from it, and that they were voluntary."
Romney's current plan for Social Security does not mention personal accounts. Instead, he proposes a gradual increase in the retirement age to account for growing life expectancy. For future generations, Romney would slow the growth of benefits "for those with higher incomes."
Romney says tax increases should be off the table and current beneficiaries and those near retirement should be spared from cuts.
"Mitt Romney and Paul support gradual reforms to Social Security that protect current beneficiaries from any benefit disruptions while strengthening the program to ensure that it doesn't go bankrupt," Romney campaign spokesman Ryan Williams said.
The trustees who oversee Social Security say the trust funds that support the program will run dry in 2033. At that point, Social Security will generate only enough tax revenue to pay about 75 percent of benefits, triggering automatic cuts unless Congress acts.
During the 2008 campaign, Obama said he wanted to improve Social Security's finances by applying the payroll tax to annual wages above $250,000. It is now limited to wages below $110,100, a level that increases with inflation.
Obama also pledged to oppose raising the retirement age or reducing annual cost-of-living adjustments, or COLAs. "Let me be clear, I will not do either," Obama said at the time.
Last year, however, Obama put on the table a proposal to reduce annual COLAs during deficit-reduction talks with House Speaker John Boehner (R., Ohio). The talks ultimately failed and nothing came of the proposal, but it raised questions about whether Obama would honor his 2008 pledge.
Obama offered some principles to strengthen Social Security in his 2011 State of the Union address.
"We must do it without putting at risk current retirees, the most vulnerable or people with disabilities, without slashing benefits for future generations and without subjecting Americans' guaranteed retirement income to the whims of the stock market," he said.
Last week, Vice President Biden made a more sweeping guarantee during a campaign swing in southern Virginia, telling a customer at a diner that Social Security will not be changed.
"I guarantee you, flat guarantee you, there will be no changes in Social Security," Biden told the customer, according to a White House pool report. "I flat guarantee you."
A Biden adviser said later the vice president was merely reassuring the woman that her benefits would not be changed. The adviser spoke on condition of anonymity because he was not authorized to speak publicly about the issue.