Originally Published: 27 JUN 21 12:10 ET

Updated: 27 JUN 21 18:36 ET

By Kevin Liptak, CNN

(CNN) -- The infrastructure deal President Joe Biden heralded this week alongside Republicans and Democrats -- and subsequently imperiled with a rogue comment afterward -- appeared back on track Sunday, even as its durability going forward remained an open question.

Republican senators, who were enraged to hear Biden deliver an ultimatum that he wouldn't sign the $1.2 trillion deal without a much larger and more partisan bill alongside, accepted his public walk-back in television appearances on Sunday morning.

A senior White House official, appearing on the same shows, still declined to say publicly that Biden would sign the deal without the separate package of Democratic priorities.

But for now, the careful political balance that seemed to be teetering has stabilized, even if the months that will pass before any bill reaches Biden's desk present ample time for further pitfalls to arise.

The episode, which unspooled from nearly the moment a deal was announced in the White House driveway, laid bare the tightrope Biden faces as he works to pass both a bill that repairs the nation's roads and bridges and a package potentially four times as large that could transform the country's economy.

The 48-hour stretch, which seemed to temporarily upend the feeling of bipartisanship Biden gleefully celebrated on Thursday, was a signal the infrastructure wrangling is far from over as lawmakers set to work making their agreement a reality.

And it was a reminder that for all of Biden's discipline as president so far, he came to the job with a well-earned reputation of uttering stray words that have the potential to derail his agenda.

"I am totally confident the President will sign it if it comes to his desk," Sen. Mitt Romney of Utah told CNN's Jake Tapper on "State of the Union" on Sunday, signaling he and other Republicans who helped broker the deal were back in the fold after balking at Biden's threat to reject the deal.

"I am glad they have now been delinked and we can move forward with a bipartisan bill," Sen. Rob Portman of Ohio, another Republican who helped secure the agreement, said on ABC. "We were glad to see them disconnected. And now we can move forward."

"I hope it's enough," Sen. Bill Cassidy, a Louisiana Republican, said of Biden's 600-word statement backing away from his vow to reject the plan if it wasn't accompanied by a much larger bill focused on climate, child care and other priorities.

"We'll see going forward," Cassidy said on NBC, "but I'll continue to work for the bill."

White House aides had spent the previous two days attempting damage control on the President's comments, which he delivered during an appearance in the East Room on Thursday after walking outside with senators to announce the agreement.

"If this is the only thing that comes to me, I'm not signing it," he said. "I'm not just signing the bipartisan bill and forgetting about the rest."

To some ears, the statement was simply a statement of what many Democrats assumed were the terms of the bipartisan deal, which left out many of the priorities Biden has championed. And it was plainly evident to both Republicans and Democrats the two bills would progress together.

But the direct linkage between the two is what angered Republicans, who complained they'd been duped into helping Biden spend trillions more than they wanted on items they don't support.

By Friday, senior White House officials who had helped secure the deal in the first place were back on the phone with senators, ensuring them Biden remained committed to the bipartisan plan they had just spent weeks negotiating.

Biden's aides sought to soften the message, and declined to repeat his assertion that he wouldn't sign one bill without the other.

Biden himself also placed phone calls to lawmakers, including Sen. Kyrsten Sinema, the centrist Arizona Democrat whose support will be required both for the bipartisan plan but also the second, larger bill likely to be passed along party lines.

But it was evident that nothing short of Biden himself publicly clarifying his stance would do. And so from retreat at Camp David, where he is spending the weekend with family, Biden issued the lengthy statement walking it all back.

"After announcing the bipartisan agreement, I indicated that I would refuse to sign the infrastructure bill if it was sent to me without my Families Plan and other priorities, including clean energy. That statement understandably upset some Republicans, who do not see the two plans as linked," Biden wrote. "My comments also created the impression that I was issuing a veto threat on the very plan I had just agreed to, which was certainly not my intent."

Biden's aides were aware that many of the Republicans who helped secure the package, and were later enraged at the President's comment, were set to appear on Sunday morning shows. On Friday afternoon, some of the senators joined a conference call to coordinate their message ahead of the broadcasts.

In addition to the statement, Biden's aides also said he would soon travel the country to help sell the plan, starting in Wisconsin on Tuesday.

"The President is focused on the historic nature of the deal and actually getting it passed, making sure it gets to his desk," said Cedric Richmond, a White House senior adviser, told Tapper later on "State of the Union." "We expect to get two bills to our desk -- to the President's desk -- so that he can sign both of them."

Still, asked whether Biden would sign the bipartisan infrastructure bill if it lands on his desk without the larger package, he demurred: "I don't think it's a yes or no question," he said.

Whether Biden's sales pitch is enough to keep both his own party behind him and to convince enough Republicans to support the smaller infrastructure plan remains unknown. Cassidy said Sunday he believed Senate Republican Leader Mitch McConnell -- with whom he is close -- would eventually get behind the bipartisan plan.

"If we can pull this off, I think Mitch will favor it," he said.

Meanwhile, Rep. Alexandria Ocasio-Cortez offered her own reminder to Biden that many liberals won't accept the bipartisan plan without the much larger bill containing the remainder of his families and jobs agenda.

"I think it's very important for the President to know that ... the Democratic caucus is here to ensure that he doesn't fail, and we're here to make sure that he is successful in making sure that we do have a larger infrastructure plan," the New York Democrat said in an interview on NBC.