WASHINGTON — The relationship between President Donald Trump and Republican leaders in Congress started as a marriage of convenience, one sustained on the promise of pushing a Republican agenda into law.

Until recently House Speaker Paul Ryan and Senate Majority Leader Mitch McConnell tolerated Trump’s turbulent debut because they agreed with the direction the White House was heading — or were confident they could nudge it in the desired direction.

Many Republicans backed the travel ban, despite the rocky rollout. They support upending Obama-era regulations and raved about Trump’s Supreme Court nominee.

But the newfound partnership is showing signs of serious strain. Growing discomfort about the Trump team’s ties to Russia, daily dramas at the White House and the increasing unrest at town hall meetings with constituents back home have prompted second thoughts about the alliance.

As Trump’s first 100 days in office tick away, and rank and file Republicans head home for a week-long recess, there is a growing worry that Congress will face a drip-drip-drip of new revelations about the Trump White House that will overshadow the rest of the Republican agenda, such as repealing Obamacare, enacting tax reform and cutting government spending.

“That’s what the fear is,” said one Republican senator, granted anonymity to frankly discuss the outlook. “It’s not a good situation. You can’t let this go and not look at it.”

In the first significant rebuke of the White House, GOP senators this week tanked Trump’s pick for Labor secretary, fast-food executive Andy Puzder.

The Senate revolt came as Trump’s firing of his national security adviser, Michael Flynn, exposed lawmakers to questions about Russian influence on the administration.

For a while, Republican leaders tried to swat back media inquiries about Trump’s latest unconventional moves or statements by refusing to engage in what they dismissed as mere distractions from the work of governing.

But now hardly a day goes by that Ryan or McConnell aren’t asked to defend Trump’s latest provocations on Twitter or the next executive order rumored to be coming from the White House.

Republican leadership is gambling that their best bet is to look past Trump’s Andrew Jackson-like coarseness in hopes of accomplishing their goals.

“There’s no question Donald Trump is a different kind of president,” McConnell said this week on the MSNBC program “Morning Joe.” “He’s different. But I like what he’s doing.”

One Republican former leadership aide said “there’s not a single Republican anywhere” who’s not stunned by some of Trump’s comments. But they focus instead on the Republican priorities they see taking shape, he said.

“In the end, we’re still talking about tax reform, Supreme Court, all the stuff is getting done,” the aide said.

Republicans have put their trust in Vice President Mike Pence, the Cabinet secretaries and a legislative team culled from the halls of Congress — even though it is unclear how much sway those voices ultimately have with the occupant of the Oval Office.

And areas of significant disagreement with Trump still loom, such as his $1 trillion infrastructure plan, having Congress pony up funds for the border wall with Mexico and a massive military buildup.

But the Russia questions are threatening to overshadow GOP goals as emboldened Democrats call for independent inquiries into alleged contacts between Trump’s campaign team and Russian intelligence officials.

A growing number of top Republicans, including Sen. John McCain, R-Ariz., and Sen. Lindsey Graham, R-S.C., are also calling for a deeper and more transparent dive into Russia’s role in the November election. That could take weeks, or more likely months.

lisa.mascaro@latimes.com