Falcons owner Arthur Blank told reporters that Green was cut Thursday morning after the team learned of the allegations the night before.
"We don't want anybody connected to the organization who has those kinds of accusations around them," Blank said.
The Salt Lake City Tribune reported that four separate sexual assault allegations were made against Green -- three to the Logan City Police Department and one to the North Park Police Department -- in 2015 while he was attending Utah State.
The Tribune reported the Cache County Attorney's Office in Utah actively investigated those allegations against Green. Capt. Curtis Hooley of the Logan City Police Department told ESPN on Thursday that Green hasn't been charged with a crime, and that Cache County Attorney James Swink declined to prosecute two of the three cases investigated by his department.
"We were not able to get enough evidence to charge him," said Hooley, who said Green was interviewed by the police officer who investigated all three cases.
A third case is still open while police investigate and speak to witnesses who were friends of the woman.
The fourth case was investigated by the nearby North Park Police Department. North Park Police Chief Kim Hawkes confirmed Thursday that his department had a sexual assault case involving Green but declined to say whether the investigation was closed or what became of it.
Cache County Deputy Attorney Barb Lachmar told the Tribune that the office is re-examining the cases and "doing some additional investigation to assist us in making our final decision."
Three of the women making the allegations against Green were students at Utah State at the time of the alleged assaults. According to the Tribune, all three notified officials at Utah State's Sexual Assault and Anti-Violence Information office before contacting police.
Green was not named in the Tribune's story last month but he denied the allegations in a July interview with the paper and said "everyone makes mistakes."
"This isn't just Salt Lake, this isn't just Logan, this isn't just Utah, the whole U.S. is about to hear about something that's blown out of proportion basically and that's going to ruin a young man's career," Green told the Tribune in remarks published Thursday.
Green was not on the practice field Thursday morning, and the Falcons announced the transaction shortly after practice began. He was highly unlikely to make the 53-man roster, although it was possible he could have remained with the team as part of the practice squad.
ESPN Staff Writers Vaughn McClure, Mark Schlabach and The Associated Press contributed to this report.