The Mirror says the former England captain, 44, quit after less than a year in charge having lifted the club off the bottom - helped by Andy Cole - and saved them from relegation.
He, his staff and his players had been battling a financial crisis at Moss Rose.
A club spokesman said last night: “We have released a statement on our website, which is the only comment that we will be making at this time."
Campbell had kept the Silkmen up thanks to a run of just two defeats in their final 10 games.
He said on Thursday: “Macclesfield Town gave me a great beginning in the managerial side of the game and fantastic learning curve which I am truly grateful for."