President Bush continued on his western swing Tuesday, where he is focusing his message about strengthening Social Security (search) on the one generation of Americans he has exempted from his plan: those in or near retirement.

Traveling to a senior center in New Mexico, Bush said rumors that seniors would not get their Social Security payments are false and misleading, but younger voters may see that rumor come true.

"The truth is you're going to get your checks. The question is whether or not your grandchildren are going to get their checks," Bush told a chuckling audience.

Critics have raised fears among many that retirees will be affected by the president's proposed reforms, damage the president is trying to undo on a second day of touring with a former GOP nemesis.

"I say to our Democrat friends, come and sit down at the table and let us work together to save the safety net for future generations of Americans," Arizona Sen. John McCain (search), known as the Democrats' favorite Republican for so-called "straight talk" as well as frequent disputes with Bush, told seniors in Arizona on Monday.

"The door is open to the White House and to the Republican side of the aisle," McCain said Monday. McCain followed Bush to events in New Mexico on Tuesday.

The emphasis on future generations is key to pushing the call for personal savings accounts. The president continued to stress the Republican approach of explaining to seniors that Social Security reform and personal accounts will benefit their grandchildren.

To make the case, last week Bush tapped his mother's credibility. Senior administration officials say they hope McCain's credibility will provide similar appeal, particularly with independents and Democratic swing voters.

"Private savings accounts work. They have been proven to work, not only in America but all over the world, and we ought to really strongly support it," McCain told the Tucson audience.

In a rare town hall-style appearance on the same day, Vice President Dick Cheney (search) repeatedly told seniors on Monday that their benefits will not be affected and they should think about their families' futures.

"This isn't about you. It's about your kids and your grandkids," Cheney said. "The focus in this debate needs to be on those younger generations." Cheney was taking that message to Reno, Nev. on Tuesday.

To combat weak polls and claims by Democratic critics that the president's Social Security agenda is already dead in Congress, Cheney appeared Monday with Rep. Bill Thomas (search), chairman of the House Ways and Means Committee, which is tasked with writing the actual legislation. Thomas pledged to pass a House bill this year.

"If you are denying there is a problem, then you really are part of the problem," said Thomas, R-Calif.

During this congressional recess, Republicans plan nearly 500 Social Security reform events across the country, an unprecedented public relations collaboration between the White House, the Republican National Committee and GOP House and Senate members. The goal is to convince voters that legislation must be passed now.

"This is an intergenerational discussion about tomorrow, and I've always found that tomorrow comes very quickly," Thomas said.

Some GOP strategists say this could be a critical week for the Social Security push because lawmakers are not in Washington talking to TV cameras but back at home for Easter recess talking to voters. During the break, Democrats too have planned a series of events critizing the president's proposal.

The GOP in recent weeks has conducted several polls and focus groups designed to help them hone a three-part message: The time for reform is now, seniors face no cuts and owe it to their kids to support reform and Democrats are obstructing the way.

Click in the box near the top of the story to watch a report by FOX News' Carl Cameron.