Auckland police said a man was placed under arrest but not charged following a "minor disorder".
According to reports, the incident occurred while Sinckler, 24, was out in central Auckland following the Lions' 1-1 series draw with New Zealand.
"I apologise for putting myself and the Lions in this position," he said.
"I also apologise to the police and anyone else affected."
Lions tour manager John Spencer confirmed the Harlequins tight-head had apologised and been reminded of his responsibilities.
Spencer said in a statement: "We have been informed by Auckland police about an incident involving Kyle.
"I have spoken to Kyle and reminded him of his responsibilities as a Lion, which extend to his off-the-field behaviour.
"Kyle has apologised for any inconvenience caused and we are satisfied that he regrets this incident and that this is the end of the matter."
An Auckland police spokesperson said: "Police can confirm they attended an incident involving minor disorder in central Auckland early on Sunday morning.
"A male was placed under arrest but was not charged after further inquiries established that the incident did not warrant prosecution.
"He was then taken to his accommodation by police. Auckland police has no further comment on this incident."
British and Irish Lions prop Kyle Sinckler has apologised after he was arrested during a night out in Auckland in the early hours of Sunday.