According to a report in The Sun, Walker allegedly hosted a house party involving two sex workers.
City will investigate the report, with Walker quick to apologise for his actions.
He said: "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 about my private life in a tabloid newspaper.
"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 re-iterate the message: Stay home, stay safe."
City also released a statement, which 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."
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