President Trump has dubbed him “Liddle Bob Corker” on Twitter, but the Tennessee Republican still wields big-time influence over American foreign policy.

While the recent criticism of Mr. Trump from Mr. Corker and from Sen. Jeff Flake — both longtime moderate Republicans and both on their way out at the end of next year — might seem like garden-variety theatrics from two lame-duck lawmakers, Capitol Hill sources say the implications are deep.

Mr. Corker heads the Senate Foreign Relations Committee, of which Mr. Flake is a member, and has suggested he intends to use the panel as a forum in the coming months to air concerns about Mr. Trump’s leadership on a range of fronts, including Iran, North Korea, Russia and the Middle East.

“The committee is going to be very active,” Mr. Corker told reporters last week before announcing plans to hold a series of hearings, which could generate headlines that damage the public’s view of Mr. Trump.

“It’s going to be a very robust period of time,” Mr. Corker said. And it all starts Monday evening, when Secretary of State Rex W. Tillerson and Defense Secretary James Mattis appear before the committee for questioning over the administration’s “perspective” on war powers and whether Congress ought to revise the Authorization for Use of Military Force, which was passed in the wake of the 9/11 terrorist attacks.

“I think it will be very informative to the American people and to the rest of the Senate about what powers the president has, should [and] shouldn’t have,” said Mr. Corker.

He also sits on the Senate Budget Committee, where his long-held anti-deficit posture could affect the president’s tax cut plan. But Mr. Corker’s comments on Mr. Trump’s military and foreign policy “powers” were particularly ominous given his chairman role on the Foreign Relations Committee.

A war of words between the men has escalated since early this month, when Mr. Trump accused Mr. Corker of retiring from the Senate because the president refused to endorse him for a third term. Mr. Corker responded by calling the White House “an adult day care center” where Mr. Trump had to be monitored closely lest he unleash incendiary tweets.

The president then referred to the senator in a tweet as “Liddle Bob Corker” after Mr. Corker told The New York Times that Mr. Trump’s conduct “would have to concern anyone who cares about our nation” and that the president’s unbridled threats to other countries could set the U.S. “on the path to World War III.”

Relations between the two men have changed dramatically since last year, when Mr. Corker was rumored to be on Mr. Trump’s short list for secretary of state nominees.

Where Mr. Corker will take the discussion next remains to be seen. Senate hearings, particularly those relating to foreign policy, tend to veer toward whatever territory a given committee chairman and his allies deem worthy.

Capitol Hill sources say Mr. Corker is likely to temper his statements in public but is poised to work behind the scenes toward pursuing his own foreign policy agenda regardless of whether it matches that of the president.

It could mean stalling Mr. Trump’s plan to reverse Washington’s commitment to the Obama-era Iran nuclear accord and pressuring the president to follow through on recently passed legislation calling for harsher sanctions against Russia.

On the North Korea front, Mr. Trump has yet to appoint an assistant secretary of state for East Asian affairs or a U.S. ambassador to South Korea. Both positions require a signoff by the Foreign Relations Committee.

Mr. Corker showed his distaste for Mr. Trump’s proposed cuts to the 2018 State Department budget, asserting in June that it would be “a waste of time” to seriously consider the cuts. In September, the Senate Appropriations Committee approved a department budget of about $51 billion, some 20 percent more than Mr. Trump wanted.

While all that hangs in the backdrop, however, Mr. Corker has claimed that his problems with Mr. Trump won’t impact the committee’s business. “My relationship with the president is not relevant,” the senator said on MSNBC last week. “I’m dealing with the principals who conduct foreign policy, and I hope that, more and more, he’ll leave these issues to them.”

Mr. Flake, meanwhile, has said neither he nor Mr. Corker will allow his criticism of the president to spill into the hearings. “Bob, he’s straight up,” Mr. Flake said, according to USA Today. “He does what he thinks is right. I’d like to think I’ll do the same.

“Our views on foreign policy are known,” the senator added. “We’re concerned about instability. And obviously, we want effective foreign policy. I think we have some issues there. But they are not borne out of animus for the president or because we disagree with the president. It’s just the way we feel.”

Mr. Flake made his comments after his own announcement last week that he will not be seeking re-election next year. The Arizona Republican stunned Washington on Tuesday with a sobering speech in which he sharply criticized Mr. Trump.

The senator accused the president of engaging in irresponsible and damaging statements, including personalized Twitter attacks against members of his own party that have amounted to a “degradation” of American politics.

“Reckless, outrageous and undignified behavior has become excused and countenanced as telling it like it is when it is actually just reckless, outrageous and undignified,” said Mr. Flake, who acknowledged that other Republicans may frown on his attack on the president.

“I’m aware that there’s a segment of my party that believes that anything short of complete and unquestioning loyalty to a president who belongs to my party is unacceptable and suspect,” he said, before adding that “the notion that one should stay silent … as the alliances and agreements that ensure the stability of the entire world are routinely threatened … is ahistoric and, I believe, profoundly misguided.”

The senator’s reference to foreign “alliances and agreements” were read by some as an indication of his readiness to take a stand against Mr. Trump on the Foreign Relations Committee.

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