His lawyer Ben Brafman - one of America's most celebrated defence attorneys - called for the entire indictment against Weinstein to be dismissed on Friday.
He was released on bail and pleaded not guilty to the charges allegedly committed against in 2004, 2006 and 2013.
Weinstein is next due to appear in court on September 20.
His arrest and arraignments have led #MeToo campaigners to hope that the twice-married father of five - whom almost 100 women have publicly accused of sexual misconduct - will be put on trial, convicted and sent to prison.
Failure by the district attorney to grant permission to exhibit the emails would warrant a dismissal of the indictment, lawyers argued.
In a motion filed in NY state court in Manhattan, Weinstein's lawyers said the emails showed "a long-term, consensual, intimate relationship" between Weinstein and an unnamed woman who has accused him of raping her in 2013.
Danish burqa ban comes into effect amid protests
It is estimated that Muslims account for about 7 percent out of the total population of 5.6 million in Denmark . A smaller demonstration arranged by "For Frihed" (For Freedom), which is pro face veil ban, also took place.
In another email from the same unnamed accuser to Weinstein on July 10, 2014 she writes, "There is no one else I would enjoy catching up with that understands me quite like you". In January 2014, the defense quoted her as writing: "Your [sic] the one who makes it look good with your smile and handsome eyes!" Just had u cross my mind and thought u would send a hello.
Weinstein is said to have replied: "Love to cross your mind it's my favorite exercise", before she answered: "Lol that made me laugh so hard". "I love you, always do".
He has denied all allegations of non-consensual sex, but he is also under investigation in Los Angeles and London. But the filings offer the most detailed look yet at how Mr. Weinstein might defend himself at trial. According to The Hollywood Reporter, the movie producer's legal team is planning to submit a motion for dismissal on the grounds that prosecutors didn't share allegedly exculpatory emails with the grand jury that returned an indictment.
If all goes to plan, the much accused and disgraced producer aims to have his criminal case attorney Benjamin Brafman file a series of motions either tonight or tomorrow in New York Supreme Court to have the grand jury indictments either dismissed or moved to another venue, Deadline has confirmed.
Last month, Manhattan District Attorney Cyrus R. Vance Jr. said the 66-year-old Weinstein is charged with "some of the most serious sexual offenses" that exist under state law.
The accusations against Weinstein led to the #MeToo movement in which hundreds of women publicly accuse powerful men in business, politics and entertainment of sexual harassment and abuse.