Walker, 29, has apologised and urged people to “stay home, stay safe” during the coronavirus pandemic.
In a statement, City said the England defender’s actions had “directly contravened” his responsibility as a role model.
It added that the club would be holding an investigation.
The government has put social distancing restrictionsin place to combat coronavirus. As of 09:00 BST on Sunday, 47,806 people had tested positive for coronavirus in the UK, with 4,934 deaths.
City said: “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.”
Last Wednesday, Walker posted on social media, urging the public to follow government guidelines on social distancing by staying indoors.