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The Sun say that police arrived at Giggs' £1.7 million house in Worsley after 10pm on Sunday following reports of "a disturbance".
Giggs was taken to Pendleton police station, where he was questioned on suspicion of actual body harm and assault.
According to the report, police say he was released on bail pending enquiries.
A spokesman for Greater Manchester Police said: "Police were called at 10.05pm on Sunday to reports of a disturbance. A woman in her 30s sustained minor injuries but did not require any treatment.
"A 46-year-old man was arrested on suspicion of section 47 assault and section 39 common assault."
In a brief statement, the Welsh FA said they are "aware of an alleged incident" involving their manager - with Giggs' press conference squad announcement for the next Wales squad cancelled.
"A statement read: "The FAW is aware of an alleged incident involving the men's national team manager Ryan Giggs.
"The FAW will be making no further comment at this current time."
As per his representative, Giggs denies all allegations and is co-operating with the police, according to the BBC.
A comment said: "Mr Giggs denies all allegations of assault made against him. He is cooperating with the police and will continue to assist them with their ongoing investigations."
Ryan Giggs has been arrested on suspicion of assaulting his girlfriend Kate Greville. pic.twitter.com/yN3PQbRq6J- Purely Football (@PurelyFootball) November 3, 2020
Giggs had been at Hotel Football, the hospitality venue he co-owns close to Old Trafford and is believed to have gone to the city centre's Stock Exchange Hotel - another venture he is involved in with former teammate Gary Neville - following United's 1-0 loss to Arsenal.
He met girlfriend Kate through the business after she started doing PR for Hotel Football. The couple have been on and off since 2017.
Giggs divorced wife Stacey in early 2018 before the relationship with Kate was made public when they were pictured on holiday in Italy.
For eight years, the 46-year-old had an affair with Natasha, wife of his brother Rhodri.
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