Potential Bush bid sways none in GOP

• Sen. Rand Paul welcomed Bush into the race: "The more the merrier!"


• A spokesman for Sen. Marco Rubio said Bush's decision would have no impact on his own


• Sen. Ted Cruz said the voters would decide on their eventual nominee

Washington (CNN) -- Former Florida Gov. Jeb Bush's surprise announcement that he'll "actively explore" a presidential bid Tuesday morning did little to dissuade his potential Republican opponents from the race.

Following the news, South Carolina Sen. Lindsey Graham, who might make a White House bid, offered the clearest pitch yet for his own candidacy.

"I think I have a unique ability to deal with the threats we face at home and abroad and the challenges here, which is finally getting the government to work and dealing with a dangerous world," he told reporters on Capitol Hill.

He said to "stay tuned" for his plans.

"I think there are a lot of people in the donor class who are looking for multiple voices, including Jeb's, and competition is a good thing," Graham said, adding, "He's got a lot to offer the Republican Party and the country."

READ: Jeb Bush to 'actively explore' presidential bid

Kentucky Sen. Rand Paul was upbeat about the governor's announcement.

"I think we're a big tent — we can use moderates, conservatives, libertarians — we need 'em all," he told reporters. "I think the more the merrier — the public will determine" whether Bush could win.

And Florida Sen. Marco Rubio's spokesman, Alex Conant, said while Rubio has "a lot of respect" for Bush and thinks he'd make "a formidable candidate," Bush's decision has no bearing on his own.

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"Marco's decision on whether to run for president or re-election will be based on where he can best achieve his agenda to restore the American Dream — not on who else might be running," Conant said.

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Many observers believe if Bush jumps in the race, Rubio — who's been mentored by the former Florida governor since his entry into politics — wouldn't run. But a source close to Rubio said Tuesday that while Tuesday's announcement changes the dynamic of the race, the junior Florida senator is still "on track to run for president" in 2016.

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Another potential opponent, Texas Sen. Ted Cruz, barely acknowledged that the two could face off in the primary fight.

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"I think what Republicans want is someone who will stand up and lead, who will take on the great challenges of the day, who will make the case that the path we're on isn't working; that the Obama-economy is a disaster, that Obamacare is a trainwreck, that our constitutional rights are under assault and that we need to restore America's leadership in the world," he said.

Sen. John Thune of South Dakota, who recently left the door open to his own 2016 bid, said the Bush announcement was "kind of expected." Now, though, "everyone is starting to feel pressure to start making moves," he said.

But even though Bush is seen as one of the top establishment picks in the primary if he runs, it was clear Tuesday morning he'll still have some work to convince some party leaders.

Arizona Sen. John McCain, the GOP's 2008 presidential nominee, wouldn't disclose what he discussed in his recent meeting with Bush, only that they went over the "ins and outs" of a presidential campaign.

READ: Source: Rubio 'on track' for 2016 run

He wouldn't back the governor, either, since he's still waiting to see whether Graham, who McCain called his "closest friend," decides to run.

Bush would enter the race with his "contributors, his family name, a long record as very successful governor," McCain said.

Sen. Rob Portman, considered a possible vice presidential pick, told reporters he has talked to Bush recently about a potential campaign.

"He'll be a strong candidate," said Portman, who recently announced he would be running for re-election rather than the White House in 2016.

None of the GOP senators noted one of Bush's biggest liabilities in a GOP presidential primary, however — his support for policies anathema to conservative Republicans. Bush's support for Common Core educational standards and his previous openness to a pathway to citizenship have drawn him many critics on the far-right, and he'll face a challenge in winning over some of those primary voters.

Soon after his announcement, the Conservative Action Fund, an outside spending group, emailed its supporters asking them to sign a petition urging Bush not to run and decrying him as "another establishment, compromising" Republican.

But even right-wing firebrand Cruz on Tuesday held his fire when given the opportunity to attack Bush on one of his primary vulnerabilities, immigration reform.

READ: 7 things to know about Jeb Bush

"Immigration will be a critical issue in this next election because we have a President who has been unwilling to work with Congress on common sense immigration reform," Cruz said, when asked how Bush's position on immigration would play out in the primaries.

And some Republicans on Capitol Hill argued Bush still had a lot of work to do simply building the operation and support he'd need to compete in a primary.

New Hampshire Sen. Kelly Ayotte, also seen as a possible vice presidential pick, knocked Bush for not yet stopping by her state — which hosts the first primary in the nation — to meet with grassroots activists.

"I don't care who you are, you have to do the hard work in New Hampshire," she said. "You gotta do the grassroots. You gotta get down there and talk to the activists."

She acknowledged that he'd have "the Bush name recognition," but said there's no substitute for hard retail politicking in the Granite State.

"It will make a difference how hard the candidates work in terms of meeting the activists and answering the hard questions," Ayotte said.