Michael Avenatti, the porn star's attorney, shared without comment BuzzFeed's report on Twitter shortly after it was published.
Michael Cohen, Trump's personal lawyer, did not respond to a request for comment.
An injunction would be the latest in a series of moves to silence Daniels, including a restraining order the president's lawyers recently obtained.
Avenatti moved the hush agreement into public view Tuesday when he filed suit on Daniels's behalf, claiming that the contract was invalid because it did not have Trump's signature.
The specter of a president who has made no secret of his hostility to the media trying to silence the porn star also raises constitutional concerns. Now that Trump is president, the existence of an extramarital relationship becomes a matter of public concern, according to C.J. Peters, dean of the University of Akron School of Law.
"A judicial 'gag order' against Ms. Daniels or CBS would constitute a 'prior restraint' of speech, which under First Amendment doctrine is almost never permissible," Peters said.
Trump's lawyers have often tried to stifle unfavorable coverage.
In 2006, one of Trump's personal lawyers, Marc Kasowitz, filed suit against biographer Timothy O'Brien, alleging that O'Brien's book "TrumpNation" underestimated Trump's net worth. A New Jersey judge dismissed that suit.
During the campaign, Trump threatened legal action against the New York Times for printing two articles about women who accused him of sexual misconduct. Both stories, Trump said, "will be part of the lawsuit we are preparing against them." A lawsuit was not filed.
In January, another lawyer representing Trump, Charles Harder, threatened author Michael Wolff with a lawsuit if he published his explosive book, "Fire and Fury." No lawsuit was filed.
The decade-old story of Daniels' alleged affair with the reality TV star was slow to reach the public eye.
Daniels, whose real name is Stephanie Clifford, told her story to InTouch magazine reporter Jordi Lippe-McGraw in a May 2011 phone interview. The interview remained unpublished in the magazine's archives. The celebrity magazine has not explained why it did not publish the story. The Associated Press has reported that four former InTouch staffers said that Trump's attorneys threatened to sue.
News of the alleged encounter appeared five months later on the Dirty, a gossip website.
InTouch published a 5,500-word transcript of the interview earlier this year, after the Wall Street Journal broke news of the hush agreement.
Since filing suit, Los Angeles-based Avenatti has attempted to win public support for Daniels, claiming that his goal is to "shed light" on her story.
Karen Tynan, a San Francisco-based lawyer who has adult-industry clients, said she thought the president's lawyers have few options to prevent the interview from airing, none of them good.
"How does it look, not only legally but politically?" Tynan said. "If they succeed it is a prior restraint of speech. If they fail, they look like they lost."
