Senate Majority Leader Mitch McConnell (R-Ky.) talks with farmer Mike Back at Mahan Farms in Paris, Ky., on Monday. (Adam Beam/Associated Press)

Senate Republicans on Monday avoided weighing in on the fiery fracas between President Trump and Sen. Bob Corker, and aides and allies of those lawmakers privately worried that a prolonged fight would hurt the GOP’s already threatened legislative priorities.

A day after Corker (R-Tenn.) and Trump traded some of the sharpest intraparty blows of the year, Republican senators were mostly quiet. Those who did speak did so obliquely — by praising Corker generally but steering clear of inserting themselves directly into the brutal clash.

“Senator Corker is a valuable member of the Senate Republican caucus and he’s also on the Budget Committee and a particularly important player as we move to the floor on the budget next week,” said Majority Leader Mitch McConnell (R-Ky.), walking a line that other Republican senators followed throughout the day. McConnell’s remarks were first reported by the Associated Press and confirmed by his office.

The reaction highlighted the broader strategy Capitol Hill Republicans have adopted when it comes to the president’s tendency to wage rhetorical war against their own or incite other controversies: Don’t engage in public no matter how anxious they may be in private.

That approach grows riskier with each passing crisis — exposing congressional Republicans to culpability for the actions, some with potentially grave global consequences, of an unpredictable and contentious president.

“They should prepare to be the ones who shoulder the blame if Trump does something truly, absolutely catastrophic,” said Rick Wilson, a Republican strategist and vocal Trump critic. He later added: “They forgot what moral courage looks like.”

Corker, who is the chairman of the Senate Foreign Relations Committee, warned Sunday in an interview with the New York Times that Trump’s bellicose threats against other nations could put the United States “on the path to World War III.” His comments came after Trump wrote on Twitter that he expected Corker to be a “negative voice and stand in the way of our great agenda.”

Neither man spoke about the other on Monday. The two have clashed increasingly in recent months on a variety of topics.

With the Senate away all week, lawmakers in the upper chamber of Congress fanned out across the country Monday. Sen. Lindsey O. Graham (R-S.C.), who has also frequently sparred with Trump, spent part of his day playing golf with the president at Trump’s private course in Northern Virginia. Afterward, Graham tweeted that he “really enjoyed” it.

In Kentucky, McConnell appeared in public with Scott Pruitt, Trump’s Environmental Protection Agency administrator, calling him an “absolutely inspired appointment.”

Pressed on Corker, McConnell repeated that the senator from Tennessee is an important lawmaker.

The public airing of the Corker-Trump dispute was enough to prompt McConnell to phone his colleague on Sunday, according to a person familiar with the call granted anonymity to reveal a private chat. Aides to both senators declined to discuss the content of the conversation.

Some Trump administration officials sent warning shots at Corker on Monday. Vice President Pence issued a statement that “no amount of criticism at home can diminish” Trump’s accomplishments on the international stage. Pence did not name Corker. White House counselor Kellyanne Conway said on Fox News that tweets such as the one Corker wrote Sunday criticizing Trump are “incredibly irresponsible.”

Like McConnell, other Republican senators aligned with Corker preferred a less hostile response.

Sen. Lamar Alexander (R-Tenn.), one of Corker’s closest allies, vouched for his colleague in a brief written statement that, to someone unfamiliar with the Corker-Trump clash, would continue to leave them in the dark.

“I work with Bob Corker nearly every day. He is a terrific United States senator, and I’m disappointed he’s decided not to seek reelection,” Alexander said.

Sen. Rob Portman (R-Ohio) went a bit further in his own statement — but without any antagonism toward the president.

"Bob Corker has been a leader in Congress on issues as diverse as deficit reduction and combating terrorism, and he is a man of unwavering integrity," Portman said. "If we're going to accomplish our economic and national security agenda we're going to have to work together, period."

Some Senate GOP aides expressed a sense of worry and resignation — worry that the Corker-Trump fight, if broadened, could threaten their effort to rewrite the nation’s tax laws, and a feeling that there was little they could do to counter the president’s unpredictability in a constructive way.

“The primary concern at this point is for the agenda,” said one Senate aide, who was granted anonymity to speak candidly. The aide added: “There is some resignation that, you know, the president is just going to throw these bombs.”

For Republicans like Wilson, who want to use every last available tactic to fight Trump, that is an unsatisfactory stance. He urged GOP leaders to be more forceful, to use congressional hearings, public comments and other tactics such as refusing to move ahead on executive branch nominees and other business, as ways to pressure the president when he does things they think are out of line.

There is a fear on Capitol Hill that such moves would trigger an all-out GOP war that would leave them with little chance to pass the laws they have long sought and have used to justify collaborating with Trump. For now, that is the winning argument.

The president and congressional Republicans have embarked on an ambitious rewrite of the nation’s tax laws they hope will finally give them a signature legislative achievement. The effort comes on the heels of a failed, months-long push to dismantle and replace the Affordable Care Act.

The collapse of that effort left a seven-year GOP promise unfulfilled and raised the stakes for a tax overhaul. With the midterm elections looming, a growing number of Republicans see it as their last chance to notch a major victory they can campaign on next year.

The tax endeavor has already been complicated by disagreements between Republicans — the same force that derailed the health-care push. GOP officials on Capitol Hill are also concerned that other battles in the party could make it even more difficult.

Some of that discord has been driven by Corker, who is not running for reelection next year. Some Republicans think the decision frees him to be more outspoken than most GOP senators about Trump — and more difficult to read in the tax talks. Corker has said he is opposed to any tax plan that "adds a penny to the deficit."

For other congressional Republicans, however, the preferred public position is to avoid the appearance of siding against the president. Sen. James E. Risch (R-Idaho), who serves on the Foreign Relations panel with Corker, offered a typical statement Monday along those lines.

“Senator Risch knows both Senator Corker and President Trump very well,” said Kaylin Minton, Risch’s communications director. “He works with both of them. Senator Corker and the president obviously have differences they need to resolve, but Senator Risch has no intention of getting involved in this matter.”

Paul Kane, Karoun Demirjian and Ashley Parker contributed to this report.