In doing so, however, Mr. Johnson brought to light new information that was ultimately unhelpful to Mr. Trump: that the American ambassador to the European Union, Gordon Sondland, had told the lawmaker that aid to Ukraine was tied to Mr. Trump’s request to have Kiev investigate Democrats. He later told reporters at a constituent stop that he tried to get permission from Mr. Trump to tell Ukraine’s president that American aid was on its way in the wake of those allegations, but Mr. Trump refused, The Milwaukee Journal Sentinel reported.

Yet a small minority of Republicans spoke out against Mr. Trump on Friday. Senator Mitt Romney, Republican of Utah and a former presidential candidate, delivered the sharpest rebuke. The president’s appeal to China and Ukraine to investigate the Bidens was “brazen and unprecedented,” Mr. Romney said, calling the conduct “wrong and appalling.”

Representative Will Hurd of Texas, a former C.I.A. officer and member of the Intelligence Committee who is retiring next year, also denounced Mr. Trump’s suggestion that China should investigate the Bidens. But in an interview on Friday with CNN, Mr. Hurd declined to make a definitive judgment about the text messages.

“I think some of these things are indeed damning,” he said. “However, I want to make sure we get through this entire investigation before coming to some kind of conclusion.”

None of those reactions are in line with the one that Republican leaders have settled on for defending the president. They have argued that there was nothing improper about the president’s suggestion that a foreign country should investigate one of his political rivals, and that no quid pro quo was suggested.

Representative Steve Scalise of Louisiana, the No. 2 Republican, said in an interview on Thursday that “a lot of people” want to get to the bottom of the rumors about the Bidens and that Mr. Trump “is echoing what people have been calling for, for a long time.” Senator Marco Rubio of Florida suggested Mr. Trump was simply trying to provoke outrage from the news media, arguing of his public appeals to China and Ukraine, “That’s not a real request.”

The party’s scattered responses underscore the challenge for Republican leaders of setting a message for a set of developments that are out of their control, said Antonia Ferrier, the former communications director for Senator Mitch McConnell, Republican of Kentucky and the majority leader.