"I want to take this opportunity to issue a public apology for the choices I made last week which have resulted in a story today [Sunday] about my private life in a tabloid newspaper," Walker said in a statement.
"I understand that my position as a professional footballer brings the responsibility of being a role model. As such, I want to apologise to my family, friends, football club, supporters and the public for letting them down.
"There are heroes out there making a vital difference to society at the moment, and I have been keen to help support and highlight their amazing sacrifices and life-saving work over the past week.
"My actions in this matter are in direct contrast to what I should have been doing regarding the lockdown. And I want to reiterate the message: 'Stay home, stay safe'."
City have expressed their disappointment and said they would investigate.
"Manchester City FC are aware of a story in a tabloid newspaper regarding the private life of Kyle Walker in relation to a breach of the UK lockdown and social distancing rules," a club statement read.
"Footballers are global role models, and our staff and players have been working to support the incredible efforts of the NHS and other key workers in fighting the effects of the Covid-19 coronavirus, in any way we can. Kyle's actions in this matter have directly contravened these efforts.
"We are disappointed to hear the allegations, note Kyle's swift statement and apology, and will be conducting an internal disciplinary procedure in the coming days."
Walker is the second high-profile Premier League player to have been caught flouting the government's guidelines after Jack Grealish.
The Aston Villa captain said he was "deeply embarrassed" after attending a late night party at the flat of a former team-mate last weekend, just hours after posting a video on social media urging fans to stay safe at home.
Grealish is being fined for the incident with the proceeds going to the University Hospitals Charity in Birmingham.
