Sen. Bob Corker, R-Tenn., talks with reporters in the Capitol's Senate subway before the Senate Policy luncheons on June 19, 2018.

A handful of Republican senators are pushing this week to curb President Donald Trump's ability to impose tariffs, after an earlier effort to do so fizzled out in the chamber.

The efforts underscore the animosity toward the president's trade policy, and its potential drag on the U.S. economy, bubbling within the Senate GOP caucus. Numerous Republicans have criticized Trump's recent string of tariffs in part because countries, in retaliating, have targeted politically important industries such as agriculture in Midwestern states.

GOP Sens. Bob Corker of Tennessee and Pat Toomey of Pennsylvania have crafted an amendment to the Senate farm bill that would give Congress the power to approve or shoot down tariffs imposed with a national security justification, according to Politico. The chamber will start considering the bill this week.

Trump opposes the amendment, and it is unclear whether it could get through the GOP-controlled Senate.

The Trump administration used the national security reasoning to impose tariffs of 25 percent and 10 percent, respectively, on steel and aluminum imports. The U.S. imposed those duties on allies including Canada, Mexico and the European Union.

All of those entities have fired back with either proposed or already levied tariffs on U.S. goods. China has responded to separate tariffs with measures of its own targeting U.S. products.

In addition, the administration is considering whether to put tariffs on automobiles and auto parts coming from the EU. The move could raise costs for U.S. consumers. Corker told Politico that such duties could cause a "jailbreak" among Senate Republicans.

"There's going to be more and more pain accruing. July 1, there's another round of tariffs on us. July 6, there's another round of tariffs on us. And we're already seeing the impact on the economy," he said on Tuesday, according to the news outlet. "This is a farm bill, if you look at the impact on the ag community of these tariffs, it is severe. So I don't know of a more appropriate piece of legislation to put this on."

Meanwhile, Sen. Jeff Flake, an Arizona Republican and frequent Trump critic, has threatened to hold up the president's judicial nominees until the Senate votes on limiting Trump's ability to levy tariffs. The senator sits on the Judiciary Committee, a panel on which the GOP holds one more seat than Democrats. Therefore, Flake could hold up judicial nominees if he chooses.

On Tuesday, he told The Daily Beast that he is "committed to getting a vote on tariffs."

Neither Corker nor Flake's offices immediately responded to CNBC's requests to comment.

Early this month, Corker unveiled legislation to check Trump's ability to impose tariffs. At the time, Trump called the senator to urge him to back off the plan, which he did not do.

Corker's effort was eventually stifled in the Senate. He then accused the "powers that be" for stopping the measure from advancing.