GOP Sens. Rand Paul of Kentucky, Marco Rubio of Florida and Ted Cruz of Texas all suggested that Bush’s announcement Tuesday that he will “actively” explore a presidential run would not affect their calculations as they eye 2016. Cruz went so far as to pan any relatively moderate Republican nominee that smacks of Mitt Romney, John McCain or Bob Dole.

Paul said he remains on track to decide on a White House run in the early spring, and a spokesman for Rubio said Bush could be a “formidable candidate” but that the Florida senator’s presidential aspirations won’t be swayed by “who else might be running.”

It’s also clear that Bush has major work to do to win over Republican senators hailing from early primary and caucus states. Sen. Chuck Grassley of Iowa said it had been at least four years since he’d talked to Bush and that the former Florida governor had made no moves to win him over. Sen. Tim Scott of South Carolina said he’d met with Bush several times in the recent years but not in a one-on-one situation.

“It will be very interesting to see how he is received immediately. And that will be, I think, a strong indication of where he goes from here,” said Scott, who’s planning to set up a South Carolina forum for GOP candidates during primary season.

New Hampshire Sen. Kelly Ayotte had a pretty succinct message for the presidential hopeful: Get to the Granite State, ASAP.

“He really hasn’t been to New Hampshire yet. I don’t care who you are, you’ve got to do the hard work in New Hampshire,” Ayotte said. “With the Bush name, he’ll have name recognition. But again I think everyone will get an open vetting in New Hampshire and it’ll make a difference how hard the candidates work in terms of meeting the activists.”

In an interview as he walked briskly to his Senate office after a round of votes, Paul stuck to his “more the merrier” message for the Republican Party’s messy primary season that looms. But he shrugged off the suggestion that he will need to base his decisions around how the rest of the field is shaping up, whether it’s Bush jumping in or anyone else.

“I haven’t really made the final decision to run yet. I’ve got to make that decision first,” Paul said. “If I were to run, I would run because I think I have something unique to offer. Other people wouldn’t make much difference.”

McCain, the GOP’s 2008 presidential nominee, appeared to be one of the few senators who had been in touch with Bush recently, though only to discuss the “mechanics” of a presidential campaign, not to offer support. In fact, McCain said he was holding out for a 2016 announcement from the hawkish Republican Sen. Lindsey Graham of South Carolina, who he cracked is “much more entertaining than I am.”

Several senators with connections to national donors praised Bush’s moves as good for the Republican Party and a welcome kick-start to the 2016 presidential race. But after interviews with 10 GOP senators, not a single one pledged to support Bush from the start. They are still waiting for the field to shape up, including several of their congressional colleagues.

Sen. Rob Portman of Ohio, who was the first rumored candidate to announce his plans (he opted against a White House run), shares many of the same pragmatic conservative bona fides with Bush. But Portman, a prolific fundraiser for the national party organs, said he would not lend his support to anyone so early.

Senate GOP Whip John Cornyn of Texas said he’s a “big fan” of Bush’s. But he noted the precarious position he’s in considering so many potential 2016ers have ties to the Lone Star State. Paul was raised in Texas, Cruz represents the state, outgoing Texas Gov. Rick Perry may also run for the presidency, and Bush has strong connections to Texas as well. His brother, former President George W. Bush, is a former governor of Texas.

“I’ve got enough to do without getting involved in that,” Cornyn said.

While most GOP senators spoke of Bush with affection, Sen. Tom Coburn of Oklahoma outright dismissed both his candidacy and that of Ben Carson, a retired Maryland neurosurgeon who’s become a rising star on the right.

Coburn said the country is ready for someone who knows “what the heck they were doing and had experience and none of it was in government” — a description that doesn’t include Bush or Carson in his eyes.

“I don’t think the country will vote for a third Bush. It doesn’t mean I don’t like the guy,” Coburn told reporters while grumbling about his disdain for presidential politics. “I wouldn’t vote for Ben Carson.”

After being asked to revive the weekend’s drama started by Cruz and Sen. Mike Lee of Utah that elicited a rare Saturday Senate session, Senate GOP Leader Mitch McConnell of Kentucky seemed relieved to be asked of Bush’s prospects rather than address the awkward internal Republican Conference politics.

Though McConnell has made clear that he will stand with Paul after the junior Kentucky senator did so much to get him reelected to the Senate this fall, the GOP leader did have some advice for Bush: Enjoy yourself while you still can.

“I always tell all of the candidates for president that the best day they’ll have will be the day before they announce. There isn’t anything harder than running for president, unless it’s being in combat and being shot at with real bullets,” McConnell said.

Seung Min Kim and Katie Glueck contributed to this report.