Al Fayed has begun the process of ending his 16-year ownership of the club and has held discussions with at least one interested buyer.
Shahid Khan, the boss of NFL franchise Jacksonville Jaguars, is reported to be among the potential purchasers.
The Pakistani-born tycoon is ranked 179th in the Forbes 400 list of richest Americans with an estimated worth of £1.65 billion.
The 62-year-old, who made his fortune as a car parts manufacturer, already has ties to London, with the Jaguars scheduled to play one NFL regular season game a season at Wembley for the next four years.
Sky Sports News contacted Craven Cottage and Al Fayed's representatives for an official response but both declined to comment, and there has been no word either from the Jaguars.
Al Fayed converted outstanding loans to Fulham into equity in January, leaving them effectively debt-free, while a second successive profit was reported for the year ending June 2012.
They have become an established Premier League club under the 84-year-old's ownership, and the forthcoming season will be their 13th straight campaign in the division.
There are also plans to revamp the club off the pitch, with planning permission in place to increase the Cottage's capacity to 30,000.
Football finance expert David Buik told Sky Sports News he believes it needs someone with vision and foresight to take Fulham to the next level.
"The new owner has got be very innovative, have a very broad vision of how he's going to make something of it," he said.
"I believe it can be done. It's a fabulous area in London, it can cope with an awful lot more spectators and we can make an awful lot of more of the brand. But it needs vision, foresight - and very deep pockets.
"If you have been going to Craven Cottage for 16 years, as I have, and been biting your nails waiting for them to get to 40 points, that's not what he wants them for.
"Unless Mr Fayed falls out with Mr Khan, these negotiations could be done by the end of the week.
"There's nothing particularily outstanding because the debt has been paid off. The key to the deal is the conversation with Hammersmith and Fulham council. They will want to know, 'What can you do for us, what can you do for the community?'"