Gov. Matt Bevin of Kentucky was unrestrained in his praise for President Trump: Opening for him at a rally on Monday, Mr. Bevin, a conservative Republican, echoed Mr. Trump’s “America First” slogan and only gently noted the nagging divisions in their party.

“We now have a president and a Congress that are united in party, and yet we still have disagreements among us,” Mr. Bevin said, insisting, “This is healthy and good.”

In private, Mr. Bevin has been blunter about the party’s disagreements. Just days before appearing with Mr. Trump in Louisville, he joined a conference call with the president’s budget director, Mick Mulvaney, to protest a White House proposal to defund the Appalachian Regional Commission, an economic development agency that spans 13 states and steers millions of dollars in federal money to Kentucky.

Mr. Bevin was not alone in his dismay.

As Mr. Trump and his advisers press for bone-deep cuts to the federal budget, Republican governors have rapidly emerged as an influential bloc of opposition. They have complained to the White House about reductions they see as harmful or arbitrary, and they plan to pressure members of Congress from their states to oppose them.