Downing arrested on suspicion of assault
The 27-year-old, who has 32 caps for England, was involved in an alleged confrontation with a woman in a pub in Yarm, in the north east of England early on Sunday morning.
"Police were called to an incident at a pub on Yarm High Street shortly before 1am on Sunday, January 8," Cleveland police said in a statement.
"A 27-year-old man has been arrested on suspicion of assault and a 32-year-old woman has also been arrested on suspicion of assault.
"Both are in custody and inquiries are ongoing."
Liverpool had no comment on the reports.
The arrest comes at a bad time for the Premier League club who are still dealing with the fall-out from two high-profile incidents of alleged racist abuse.
Forward Luis Suarez was banned for eight games by the FA for abusing Manchester United's Patrice Evra while Merseyside police arrested a man on Saturday in connection with alleged abuse during Friday's FA Cup victory over Oldham Athletic.