Staveley's investment fund PCP Capital Partners are in talks with the current Magpies owner.
She is backed by £28bn from the Middle East and China and the Daily Mail claim a deal could be done by Christmas.
The reports also say Rafa Benitez could be handed £300m to spend if Staveley completes her takeover.
Ashley bought Newcastle for £134m back in 2007 but has been unpopular with the fans.
Staveley was also key when Sheik Mansour bought Manchester City in 2008.
Ashley is understood to have been approached by four bidders and is hoping that can drive his selling price up.
The Newcastle owner's lawyer, Andrew Henderson, said: "Our intentions at the moment are to see if the club can be brought forward into new ownership by potentially Christmas.
"Assuming that an appropriate person can be found to take the club forward, on exactly how payments are structured, how much would be payable day one, how much might be payable further down the line, there is a real genuine desire from our point of view to make sure that the club is put in the best possible position going forward.
"So if that means us helping with additional funds being available in the short term, we would seek to accommodate that."
Meanwhile, a club statement read: “As one of the Premier League's oldest and best supported football clubs - and for the benefit of its many fans and supporters in the UK and across the world - Newcastle United requires a clear direction and a path to a bright and successful future.
“To give the club the best possible opportunity of securing the positioning and investment necessary to take it to the next level, at what is an important time in its history, its present ownership has determined that it is in the best interests of Newcastle United and its fans for the club to be put up for sale.
“To give an incoming owner the maximum possible flexibility to make meaningful investment in the club, including in its playing squad, the sale process will give interested parties the opportunity of deferring substantial payments.
“A process will now commence of identifying new ownership for the club that will be capable of delivering the sustained investment in and dedication to the club that is necessary for it to achieve its ambitions.”
Newcastle return to Premier League action on Saturday when they host Crystal Palace.
