WASHINGTON—Republican leaders and donors, who for months took a frosty, hands-off view of Donald Trump, this week significantly warmed to the New York real-estate mogul as the fractured party began closing ranks behind its presumptive presidential nominee.

Beginning with his visit with GOP congressional leaders on Capitol Hill late last week, Mr. Trump has made steady inroads with the party’s elite as prominent skeptics have dropped or buried their objections in deference to their shared quest to defeat likely Democratic nominee...