(CNN) Pro-free trade Republican Sen. Jeff Flake of Arizona formally introduced legislation Monday to nullify the tariffs on aluminum and steel that President Donald Trump announced last week. But several senior GOP senators, who also are opposed to the tariffs, said a legislative response may not be necessary now that Trump has indicated he's open to exemptions for several key countries.

"I think we're making progress without legislation," said Sen. John Cornyn of Texas, the second-ranking Senate Republican. "I think the President carving out states that aren't hostile actors when it comes to trade is a positive development, and I expect we'll continue that conversation."

"Let's wait and see what happens," said Senate Finance Committee Chairman Orrin Hatch, R-Utah, whose committee has jurisdiction over the issue. "I think the President knows that we're going to have to work that out."

Sen. John Thune of South Dakota, the third-ranking GOP senator and the chairman of the Commerce Committee, has been a vigorous opponent of the tariffs. He was asked if the sentiment in Congress was shifting away from legislation to turn back the protectionist penalties against foreign producers.

"I think we're probably going to pay a little bit of attention to what the administration does and how they implement this going forward," Thune said. "If they're taking Canada and Mexico out, which is a move in the right direction, the question is, what about European allies, etc."

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