(CNN) Patience is beginning to wear thin among Senate Republicans for a prolonged government shutdown, with a growing number at risk of revolting if the fight over funding President Donald Trump's border wall drags out indefinitely.

In interviews with a wide range of Senate Republicans, a number have grown anxious about the long-term impact of a partial shutdown that has shuttered a quarter of the federal government and impacted hundreds of thousands of federal employees and contractors, including many in their states who will soon miss their first paychecks. And some GOP senators are signaling a new willingness to allow a number of key federal agencies -- such as the Treasury Department, the Transportation Department and the National Park Service -- to reopen even if the fight over the border wall is not resolved, breaking with the White House and Senate GOP leaders.

"I think we can walk and chew gum," GOP Sen. Lisa Murkowski of Alaska said on Tuesday, adding she's "amenable to a process that would allow for those appropriations bills that have concluded some time ago that they be enacted into law -- whether it's the Department of Interior or the IRS. I'd like to see that."

In a warning sign for the White House, a number of Senate Republicans offered tepid support for the hard-line stance taken by Trump, who is backed by Senate Majority Leader Mitch McConnell. While many Republicans said they'd continue to support McConnell and Trump for now, it was clear that support won't last indefinitely.

"There's a time when that may run out," Georgia Sen. Johnny Isakson said of support for McConnell's refusal to back spending bills that lack Trump's support. "But right now that doesn't run out."

House Democrats last week passed six full-year spending bills to reopen federal agencies impacted by the shutdown as well as a stopgap bill for the Department of Homeland Security, and they have put pressure on congressional Republicans and the administration to take up the measures despite an ongoing dispute between Democrats and the President over funding for a border wall. This week, they plan to pass individual spending bills to reopen several key agencies, including the Interior Department and Housing and Urban Development.

Trump, however, flatly opposes reopening other agencies without securing his demand for $5.7 billion in border wall funding -- a position backed by House and Senate GOP leaders.

But West Virginia GOP Sen. Shelley Moore Capito, who chairs a key subcommittee overseeing homeland security funding, said she "can live" with a continuing resolution to reopen the Department of Homeland Security while other agencies also are reopened. Doing so would punt the funding fight over the wall until a later time.

While Capito said she wanted to hear Trump's pitch to the American public and that "there's a real crisis" at the border, she added there's "no way" the fight would drag on for years as Trump warned last week.

"I think certainly I have expressed more than a few times the frustrations with a government shutdown and how useless it is," Capito said Tuesday. "That pressure is going to build."

Murkowsi concurred. "I am supportive of a process that is going to allow us to get these six bills through and if we need ... to do something different with Homeland ... then let's do that."

The comments come at an urgent time for Trump, who plans to attend a Senate GOP lunch Wednesday to shore up support for his strategy with the shutdown now in its third week. Already, Republican Sens. Cory Gardner of Colorado and Susan Collins of Maine -- who are both up for re-election in 2020 in purple states -- have indicated that they would support action by Congress to reopen the government even without a border wall deal. And more could soon revolt.

"Yes," Collins said directly when asked if she wanted McConnell to allow votes on individual spending bills to reopen the government.

Others signaled they could be on board as well with allowing other federal agencies to open.

Sen. Marco Rubio of Florida told CNN he would "potentially" support voting on bills to reopen specific agencies absent a border security deal, but he cautioned he would need to see the bills first and understand how they're structured.

"Potentially, I have to see the proposal," Rubio said. "It depends how it's constructed. I don't want to take away any of the President's leverage, but I don't think shutdowns are good for him -- or for us or for anyone. It's certainly not good for Americans."

Sen. Pat Roberts, the Kansas Republican who chaired the Senate Agriculture Committee, said "it could be a problem" in the long term for the Department of Agriculture if the fight is not resolved.

"Down the road it could be a problem," Roberts said.

Sen. John Boozman of Arkansas said he's concerned about families who live "paycheck to paycheck" affected by the shutdown. While he wanted to hear more from the President, Boozman added this about the federal workers caught up in the standoff: "We don't need to hold them hostage."

Asked about moving on individual appropriations bills, Montana Sen. Steve Daines said: "We should be doing everything we can here to minimize the impact of this partial shutdown. Voting on appropriations bills -- we can do that."

He also said there are other things that can be done to ease the pain, such as working with the Interior Department to tap into existing funds to pay for certain services in national parks.

"The two sides should be able to work together here and find middle ground where we can absolutely open up the government and fund border security," Daines said. "If we can't do that, we should find a different line of work."

Indeed, a number of Republican senators on Tuesday continued to express support for the President's push for increased border security and defended his request for wall funding as reasonable.

Sen. Thom Tillis of North Carolina, who is up for re-election in 2020 and viewed by Democrats as a swing vote, shut the door on opening up specific agencies as the border wall fight plays out.

"No," Tillis told CNN when asked if he would be willing to open individual agencies before the wall fight is resolved, saying he wants border security funded. "I don't want to start picking it away. We know how to get this done. We just need to execute."

Sen. Lindsey Graham, a South Carolina Republican, said: "My goal is not to reopen the government. My goal is to fix this problem the best we can. Now is the time," referring to border security.

Senate Appropriations Chairman Richard Shelby of Alabama also said he would not support moving forward on individual appropriations bills, while McConnell criticized Democrats for refusing to agree to the President's request for border wall money, saying they have put a "partisan tantrum" ahead of keeping a quarter of the government open.

Still, a number of Senate Republicans also appeared reluctant to endorse the idea of the President invoking a national emergency to secure funding for the wall -- a possibility that Trump has talked about in recent days.

Asked if it would be wise for Trump to make an emergency declaration over the wall, Ohio Sen. Rob Portman said: "No. It's a much wiser idea to negotiate something out, because if it's done it will be tied up in the courts for a long time, and it would not be successful in achieving the objective. Plus it's cumbersome."

This story has been updated to include additional comments from lawmakers later Tuesday.