Various reports in the press claim Al Fayed is in advanced talks with the Pakistan-born billionaire - and that a deal could be concluded by the end of the week.
Khan 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.
Al Fayed, who bought Fulham in 1997, has transformed them into an established Premier League side and taken them into Europe where they reached the Europa League final in 2010.
There are also plans to revamp the club off the pitch with planning permission in place to increase Craven Cottage's capacity to 30,000.
The club is also effectively debt-free after Al Fayed, 84, converted previous loans into equity earlier this year.