The offer comes as CBS News' "60 Minutes" prepares to air an interview with Daniels, whose real name is Stephanie Clifford.
Trump's lawyers reportedly have considered seeking a legal injunction to bar that interview from airing — a court order that may be impossible either to get or to sustain on appeal.
The interview could air as early as Sunday. The offer Daniels made Monday explicitly seeks Trump's agreement not to try to stop that interview from being broadcast, according to a copy of the offer published by the New York Times.
Daniels' lawyer sent a letter Monday to Trump's personal lawyer, Michael Cohen, who cut that hush-money deal, as well as to Cohen's own attorney, offering to pay Trump the $130,000, NBC News reported.
The offer is due to remain open only until noon Tuesday, according to the letter.
If Cohen accepts, Daniels could "speak openly and freely about her prior relationship with the president and the attempts to silence her and use and publish text messages, photos and videos relating to the president that she may have in her possession, all without fear of retribution or legal liability," the letter said, according to NBC News.
"This has never been about the money," Clifford's lawyer, Michael Avenatti, told NBC News. "It has always been about Ms. Clifford being allowed to tell the truth. The American people should be permitted to judge for themselves who is shooting straight with them and who is misleading them. Our offer seeks to allow this to happen."
On the eve of the 2016 presidential election, Daniels accepted $130,000 from Cohen and signed an agreement, using a pseudonym, to not talk about her alleged relationship with Trump.
A lawsuit filed last week by Daniels asks for that nondisclosure agreement to be voided on the
grounds that Trump never signed it. She claims in the suit to have had an "intimate" relationship with him from 2006 to 2007.
Melania Trump gave birth to their son, Barron, several months before the president's alleged relationship with Daniels began. Trump denies the affair, according to the White House.
Before Daniels' suit was filed, Cohen obtained a temporary restraining order barring her from talking about the "confidential information" covered by the agreement.
Daniels' lawyer claims that order is not valid.
CNBC has reached out for comment from Cohen and Avenatti.
A White House spokesman declined to comment. A spokesman for "60 Minutes" declined to comment.