New York prosecutors have subpoenaed former President Donald Trump's deposition in a lawsuit filed by the writer E. Jean Carroll for use in the state's "hush money" criminal case against him.

In a court filing Tuesday, prosecutors maintained that upon reviewing portions of Trump's video deposition in the E. Jean Carroll case that were publicly released, "a number of the subject matters about which defendant testified under oath relate to facts at issue in this case and are therefore relevant and material to this proceeding."

Trump has entered a not guilty plea to 34 felony counts of falsification of business records related to a payment his former attorney made to adult film star Stormy Daniels days before the 2016 presidential election.

Prosecutors cited Trump's statements about the "Access Hollywood" tape that were made public in October 2016, just before the presidential election and argued the testimony "features prominently in the People's case."

Prosecutors are also interested in Trump's testimony about allegations of sexual misconduct by two other women, arguing, "the way in which defendant dealt with allegations of a sexual nature by women in the months leading up to the 2016 presidential election is clearly relevant to the allegations in the People's case."

In filings that have not been made public, Trump's legal team sought to quash that subpoena and another seeking a variety of communications between Trump Organization employees and White House staff between Inauguration Day 2017 and Dec. 31, 2017. Some of the individuals listed include former Trump CFO Allan Weisselberg, Ivanka Trump, aide Dan Scavino, former Trump lawyer Michael Cohen, former Trump Organization chief legal officer and Israel adviser Jason Greenblatt, former White House communications director Hope Hicks and others.

Trump also sought to quash subpoenas over a longer period of time, from Jan. 1, 2015 through Jan. 20, 2017, of all of the emails between his wife, Melania Trump, and longtime Trump assistant Rhona Graff, as well as his travel itineraries for that period of time.

The state of New York charged Trump in April, the first time a former president had ever been indicted. The Manhattan District Attorney's Office alleges Trump orchestrated a "catch and kill" scheme to suppress damaging information before the 2016 election. State prosecutors say it involved falsifying business records to conceal three payments, including $130,000 that Cohen paid to Daniels.

Trump pleaded not guilty to all charges.

Graham Kates is an investigative reporter covering criminal justice, privacy issues and information security for CBS News Digital. Contact Graham at [email protected] or [email protected]