Several media outlets, including the BBC, said the firm had made a bid to Newcastle owner Mike Ashley.
Newcastle are yet to comment on the reports.
The bid is likely to be backed by funds from her extensive network of contacts in the Middle East, Asia and elsewhere.
Globally PCP has $37 billion of assets under management, with Staveley acting for big Gulf investors in London real estate and other assets, as well as for wealthy Chinese investors.
Staveley is believed to have been in talks with Newcastle about a potential takeover for several weeks.
Sportswear magnate Ashley announced he wanted to sell the club in October after 10 years in charge, having paid £134.4 million to buy Newcastle in 2007. It has been reported that he wants nearer £400 million for northeast side.
The 53-year-old Londoner has proved a divisive figure during his tenure, which has seen Newcastle twice relegated from the Premier League and then immediately promoted back.
Staveley, 44, helped broker Sheikh Mansour’s purchase of Manchester City in 2008.
She was spotted watching Newcastle’s 1-1 draw with Liverpool at St. James’ Park last month.
Newcastle were promoted from the Championship under Rafael Benitez last season.
The northeast club regularly attract crowds of over 50,000, but their supporters have not had a trophy to celebrate since the 1969 Inter-Cities Fairs Cup — the precursor to the UEFA Cup.