WASHINGTON — At a private conclave with the billionaire Koch brothers’ political apparatus this year, Senator Mitch McConnell of Kentucky, the Republican leader, laid out a confrontational agenda for a Republican-controlled Senate aimed at dismantling President Obama’s legislative successes through the federal budget.

In an audio recording leaked to The Undercurrent, a liberal-leaning YouTube channel, and initially reported by the magazine The Nation, Mr. McConnell told the mid-June gathering in Dana Point, Calif., that if the Republicans gained control of the Senate and retained control of the House in November, Congress could use the budget process to force the president to roll back his priorities.

“In the House and Senate, we own the budget,” he said, explaining that the initial blueprint on taxes and spending does not require the president’s signature. “So what does that mean? That means that we can pass the spending bill. And I assure you that in the spending bill, we will be pushing back against this bureaucracy by doing what’s called placing riders in the bill. No money can be spent to do this or to do that. We’re going to go after them on health care, on financial services, on the Environmental Protection Agency, across the board. All across the federal government, we’re going to go after it.”

The channel released audio of three other Republicans in tough Senate races — Representative Tom Cotton of Arkansas, Representative Cory Gardner of Colorado and Joni Ernst, a state senator in Iowa — all of whom praised Charles G. and David H. Koch and the millions of dollars they have provided to help Republican candidates.