Sussex Police Authority voluntarily referred the allegation last month.
A Sussex Police spokeswoman said: “Given the seniority of the office of Chief Constable, Sussex Police believes it is paramount that any allegations about the chief are subject to the independent scrutiny of the IPCC in order to maintain the public’s confidence in the outcome of any investigation.
“We welcome their scrutiny and will provide any support required to the investigation.”
Chief Constable Richards said: “Given the senior position I hold and the trust placed in me by the people of Sussex and my colleagues, it is absolutely right that the Sussex Police Authority has voluntarily referred this matter to the IPCC for independent scrutiny.
“I will cooperate fully with the investigation to help them establish the facts.
“With trust in the police at the forefront of ongoing legitimate public interest nationally, I am mindful of the need to conduct my duties with the utmost integrity and an expectation of scrutiny.
“This is a view shared across the force and we have recently updated all our policies in this area to underline this commitment.
“While the IPCC investigation is ongoing it will be business as usual.
“There will be no impact on the service provided to the public or the delivery of operational policing, which is our foremost priority.”
Sussex Police Authority and the IPCC have not revealed the nature of allegations against Chief Constable Richards due to the “risk of an impact on an on-going operational matter”.
“We will provide more details when there is no longer a potential impact on operational matters or when the IPCC concludes their investigation,” said the spokeswoman.