Councillor David Graham, who represents Buckhaven, Methil and Wemyss on Fife Council, has been suspended from the party while investigations are ongoing.
He has also stood down from his council appointments for the time being, according to the local authority.
Graham was the council’s spokesperson for health and social care and a member of the NHS Fife board.
He is expected to appear at Kirkcaldy Sheriff Court in connection with the alleged offence at a later date.
Scottish Labour said, as an active investigation was ongoing, it would make no formal comment on the matter.
Fife Council’s head of legal and democratic services, Lindsay Thomson, said: “Councillor Graham was charged with an offence and has been suspended by the Labour party.
“He has stepped down from his council appointments pending the outcome of the investigation. No further comment can be made by the council during a live police investigation.”
A spokesperson for NHS Fife added: “NHS Fife is aware of a Police Scotland matter involving a local Fife councillor. We are unable to comment further on an ongoing police investigation.”
A Police Scotland spokesperson said: “A 41-year-old man has been arrested and charged in relation to grooming offences following an investigation in Fife.
“He has been released on an undertaking to appear at Kirkcaldy Sheriff Court at a later date.
“Further inquiries are ongoing.”