Some party leaders remain hopeful that they can block Mr. Trump by denying him a majority of the delegates to the July convention and coalescing support around another candidate.

But the National Republican Senatorial Committee has already conducted polling to test the message that Republicans must control the Senate as a check against a President Hillary Clinton, and that Democrats must not be allowed to fully control the appointment and confirmation of Supreme Court judges, according to two people briefed on the research, who spoke on the condition of anonymity because it was intended to be confidential.

Democrats see Mr. Trump as increasing their chances, especially in diverse and fast-growing states like Arizona and Virginia, where the party often struggles to turn out Hispanic voters who can help its candidates. And the Democratic Congressional Campaign Committee is mounting a late push to stretch the political map by recruiting candidates in as many as 10 conservative-leaning House districts, in states like Florida and Kansas, where analysts believe Mr. Trump will harm Republicans.

The Democratic committee, eager to cut into the Republicans’ majority, has begun a large data project to model both support for and opposition to Mr. Trump. Meredith Kelly, a spokeswoman for the committee, confirmed that its data team was studying which of Mr. Trump’s ideas and comments would be most offensive to key voting blocs, and how best to project those themes in congressional races.

Gov. Dannel P. Malloy of Connecticut, chairman of the Democratic Governors’ Association, said Republicans had a no-win situation on their hands: They could either run from Mr. Trump and risk depressing Republican turnout, or embrace him and have to defend “views that are abhorrent” to many people.

“Quite frankly, we’re going to hold people accountable: Are you with Trump and his policies, or are you against him?” Mr. Malloy said.