Republican Party officials have pressed Mr. Trump to sign a joint fund-raising agreement, which would allow him to raise money for the national committee and for his own campaign simultaneously. That, in turn, would also give Mr. Trump a defensible answer for why, after months of railing against Wall Street executives and special interests, he recently turned to a former Goldman Sachs executive, Steven Mnuchin, to corral large checks for his campaign.

Both Mr. Trump’s aides and party officials were caught by surprise by the abrupt end of the primary contest last week, when Mr. Trump carried Indiana, prompting Senator Ted Cruz of Texas and Gov. John Kasich of Ohio to withdraw from the race. But the two sides have hurried to wrap up a joint fund-raising agreement, and one is close to being signed, according to people close to the national committee who were not authorized to speak publicly.

“As soon as there’s unity, it’s going to be very easy to do,” Mr. Trump said in an interview Monday, adding that he still planned to write checks for his campaign. “I think we’ll raise $1 billion,” he said.

Under a joint fund-raising agreement, Mr. Trump and the party would most likely be able to raise even more than the current individual limit. But such efforts are difficult and take time: While the limits were lower in 2012, Mitt Romney raised less than $500 million under such an agreement that year, using a donor network that had taken years to develop.

In one sign of progress for Mr. Trump, Stanley Hubbard, a billionaire broadcasting executive who had donated money to efforts to thwart him, fell in line behind him. “All my other candidates withdrew, one by one,” Mr. Hubbard said in an interview. “He was the last man standing.”

But other donors remain staunchly opposed. Paul Singer, the billionaire financier who had backed Senator Marco Rubio of Florida, made clear at a gala Monday night for the Manhattan Institute, a conservative think tank, that he could not support Mr. Trump and was dismayed by the likely choices in the general election.