The Manhattan district attorney's office will start presenting its evidence in the case on Monday, the newspaper reported, which would lay the groundwork for any criminal charges against Trump.
A witness in the case, former National Enquirer publisher David Pecker, was seen entering the lower Manhattan building where the grand jury is empanelled, the Times reported. The publisher had offered to help Trump by buying rights to unflattering stories and never publishing them.
The moves are an indication that the district attorney, Alvin Bragg, is closer to a decision on whether to charge Trump.
Bragg's office declined to comment on the report.
Daniels said she had a sexual liaison with Trump and received $US130,000 ($A183,700) before the 2016 presidential election in exchange for not discussing her encounter with Trump, who denies it happened and in 2018 told reporters he knew nothing about a payment to Daniels.
Trump's former personal lawyer, Michael Cohen, was sentenced to three years in prison in federal court in New York for orchestrating hush payments to Daniels and another woman, former Playboy model Karen McDougal, who said she had a months-long affair with Trump before he took office.
McDougal has said she sold her story for $US150,000 ($A211,960) to American Media Inc (AMI), but it was never published. The incident involved a practice known as "catch and kill" to prevent a potentially damaging article from being published.
Pecker, AMI's former chief executive officer and a longtime friend of Trump and Cohen, told prosecutors of their hush-money deals with McDougal and Daniels before the 2016 US election won by Trump, the Wall Street Journal reported in 2018.Â
