WASHINGTON—When revelations about President Trump’s efforts to pressure Ukraine first started dribbling to the public in late September and early October, Republicans were caught off guard.

Some members of the GOP criticized Mr. Trump’s request for Ukraine to open investigations that benefited him politically, while others sought to avoid the topic altogether. As many as 20 House Republicans initially were open to supporting Mr. Trump’s impeachment, according to Rep. Pete King (R., N.Y.), a retiring member from a competitive...