The 27-year-old Brazilian denies the allegation of rape and insists he is the victim of attempted blackmail, Sky Sports reported.
Reuters and Associated Press say they have seen a document filed in Sao Paulo alleging the footballer drunkenly assaulted his accuser at a hotel in Paris.
The former Barcelona player, who signed for PSG for a world-record £198 million in 2017, is alleged to have met the woman after they exchanged messages on Instagram.
His assistant is said to have sent the woman plane tickets on May 15 and the pair checked in to the hotel.
According to the claims in the police report, after Neymar and the woman "caressed", the footballer became "aggressive", then used "violence" and "practiced sexual intercourse against the will of the victim".
The player posted a seven-minute video to Instagram on Sunday in which he denied the accusation and insisted it was "a setup".
The video included images and messages which he said he exchanged with the accuser in a friendly way after the date of the alleged rape.
Neymar was called to speak with authorities on Thursday about posting images and messages of the accuser to social media without her authorization in possible violation of her online privacy.
The Brazilian arrived at the police headquarters on Thursday in a wheelchair after rupturing his ankle ligaments in a friendly against Qatar on Wednesday.
He testified for around two hours before speaking to reporters outside the station.
"I only want to thank the support and messages that everybody sent," he said.
Neymar is expected to speak to authorities in the coming days about the rape allegation, which is being investigated separately in Sao Paulo.
On Friday, Nike admitted their concerns about the accusation made against Neymar, raising questions about their sponsorship of the PSG star in the future.
The sports clothing manufacturer issued a statement which read, "We are very concerned by the recent allegations and will continue to closely monitor the situation."