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.

Of acute concern to Republicans are a handful of low-profile programs aimed at job training and economic revitalization, including regional development agencies like the Appalachian commission and the Delta Regional Authority, which serves eight Southern and Midwestern states, seven of them with Republican governors. They are also protective of grants from the Department of Housing and Urban Development and a $3.4 billion job-training program funded through the Labor Department.