Some Republican senators on Capitol Hill, including one of President Trump’s most ardent supporters, have signaled their uneasiness to the sweeping global tariffs that the president announced this week and that sent global markets reeling.

Senator Ted Cruz, Republican of Texas, warned on Friday that a future where other countries slap retaliatory tariffs on U.S. goods, as China has already done, was a “very real possibility” and would be a “terrible outcome” for the country.

“It’s terrible for America,” Mr. Cruz said on the latest episode of his podcast. “It would destroy jobs here at home and do real damage to the U.S. economy if we had tariffs everywhere.”

Mr. Cruz also said that a trade war would likely push inflation up and burden consumers with higher costs.

“I love President Trump. I’m his strongest supporter in the Senate,” Mr. Cruz said. “But here’s one thing to understand: A tariff is a tax, and it is a tax principally on American consumers.”

Mr. Cruz’s comments came just two days after the Senate, in a largely symbolic move, voted to halt planned 25 percent levies on Canada. However, the bill is almost certain to die in the House — and even if it does not, Mr. Trump would be unlikely to sign it.

Mr. Cruz was not among the Republican senators who joined all Democrats in pushing the bill through. They were Senator Lisa Murkowski of Alaska, Senator Susan Collins of Maine, and Senators Rand Paul and Mitch McConnell, both of Kentucky.

On Thursday, another top Republican senator — Chuck Grassley of Iowa — teamed up with a Democrat to introduce a bill aiming to reclaim congressional authority over the implementation of tariffs.

The bill, which Mr. Grassley co-sponsored with Senator Maria Cantwell, Democrat of Washington, would require the president to give Congress 48 hours notice of any new tariffs. Congress would then have to approve those tariffs within 60 days, or they would expire. Mr. Cruz was not a co-sponsor.