City said it was "disappointed" after Sunday newspaper claims that Walker hosted a party with a friend and two women at his home in Cheshire.
The club said Walker's behaviour "directly contravened" its efforts to support key public services during the pandemic, and said he would face a disciplinary investigation.
Walker's alleged breach of social distancing guidelines comes after a separate episode involving Aston Villa midfielder Jack Grealish, who apologised after visiting a friend last weekend.
City said in a statement: "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.
"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, 29, has made 21 Premier League appearances during the current campaign, his third season at City.
The England full-back won back-to-back titles during his first two seasons after joining the club from Tottenham.
He expressed regret for stepping out of line amid the allegations against him.
"I want to take this opportunity to issue a public apology for the choices I made last week," Walker told The Sun.
"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."