"The ball is really in our court right now," Collins said. "Our job is not done."

The Maine Republican called for a return to the normal legislative process by addressing problems and crafting bills on the committee level before bringing them to a vote before the full chamber.

"We need to go back to committee," she said.

Collins also emphasized her opposition to more comprehensive legislation and instead advocated the Senate "produce a series of bills" aimed at addressing pressing issues in health care. She said the first issue the Senate should focus on would be to stabilize the insurance markets.

"I certainly hope the administration does not do anything in the meantime to hasten that collapse," Collins said.

Sen. Jeff Flake, a Republican senator from Arizona, said he was disappointed the proposal had failed, but echoed Collins' call for a return to the committee process in an interview on CBS's "Face the Nation."

"That'll be certainly good for us and good for the country," Flake said.

In a series of tweets over the weekend, President Donald Trump has called on the Senate to change its rules and move health care legislation through without the threat of a filibuster, meaning if a simple majority of Republicans could agree on a proposal, it could pass.

Flake expressed his opposition to Trump's call for the Senate to change its rules and said a degree of bipartisanship was needed in the chamber.

"Even if you change the rules of the Senate, which we should not do, there are limits to what one party can do," Flake said.

When Senate Majority Leader Mitch McConnell moved to eliminate the filibuster for Supreme Court candidates in April, he reiterated his opposition to eliminating the filibuster for legislation, and despite Trump's repeated tweets pushing for an altering of the rules, there appears to be little appetite in the Senate for the change.

Collins contended that the cost-sharing reduction payments to insurance companies to cut deductibles and co-pays for lower-income people were not "bailouts," and that although the payments went to insurance companies, they have helped many who need it most to gain access to health care.

"It really would be detrimental to some of the most vulnerable citizens if those payments were cut off," Collins said.

On the second half of Trump's tweet, targeting members of Congress, White House budget director Mick Mulvaney said in a separate interview on the Sunday program that Trump was eying ending employer contributions for health insurance for members of Congress.

Lawmakers get insurance under the exchange system established under Obamacare, but as Mulvaney explained, the Office of Personnel Management during the Obama administration decided members would receive the employer contribution from the federal government.

"That's the rule that the President was talking about in his tweet yesterday," Mulvaney said.

Mulvaney further clarified that Trump's tweet demanding a vote on health care before any other legislation was the White House's official position.

"In the White House's view, they can't move on in the Senate," Mulvaney said.

He argued Republicans in Congress need to pass a bill targeting Obamacare as a matter of practicality and politics.

"You promised folks you'd do this for seven years," Mulvaney said. "You cannot go back on that."

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Susan Collins: 'Go back to committee' on health care - CNN