They also were looking for any evidence that the lawyer, Michael Cohen, had tried to hide damaging information about Trump during the election, sources briefed on the warrant told the Times.
In addition, prosecutors reportedly are looking at whether Cohen committed bank fraud, along with possible improper campaign donations to the president.
In the Access Hollywood video, Trump is heard on a live microphone bragging in extremely vulgar terms about groping and kissing women without their consent.
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He further challenged the federal government to prove his claims wrong or "stop negotiating for numerous girls that don't exist". The government needed to deliver "meaningful action on behalf of all these victims of Boko Haram's crimes".
"Absent a search warrant, these records could have been deleted without record, and without recourse", prosecutors wrote.
The warrant is the first indication that investigators suspect there was any effort to suppress the tape.
The news indicates that investigators are focused on finding out more about the nature of Cohen's role during the 2016 campaign.
The New York Times reported Wednesday that the Federal Bureau of Investigation agents who searched Cohen wanted "all records" related to the tape during their Monday raid.
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Swift tends to be selective with her recorded cover songs, so any one she decides upon would be an event. Regardless of Swift's personal connection, some people did not appreciate the lyrical change.
An attorney for Cohen didn't respond immediately to request for comment. Daniels was paid $130,000 during the 2016 campaign.
Previous reports said Federal Bureau of Investigation agents in the search had sought records on payments to Stormy Daniels and to another woman who claimed she had an affair with Trump.
Trump "is the President of the United States", Hendon said.
Rumors continue to swirl whether Trump will fire Mueller or Rod Rosenstein, but so far the White House has not confirmed that. The President recently denied any knowledge of the payment Cohen made to Daniels to keep her story quiet. This news seems to support critics who believe that Special Counsel Robert Mueller - who reportedly referred the raid in part to the Southern District of NY - is really pushing his limits here, despite his claims to the contrary. There has been no subpoena, the source said. They also asked that certain documents related to the case remain sealed to protect the privacy rights of "innocent third parties" who would be subjected to a media circus if their names became public. NBC has long held that it was about to broadcast the tape when The Washington Post obtained and published it. The president has already terminated or dismissed many persons who did not tow his line of thought.
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President Trump did not specifically state the targets but did say the targets were associated with chemical weapon capabilities. Associated Press reporters in Damascus saw smoke rising from the city's east Damascus early Saturday morning local time.