The Cottagers are looking to bolster their side this summer in order to build a side that can avoid relegation from the Premier League next term.
Swansea meanwhile look set to lose a number of their high-profile players this summer following their relegation from the top-flight.
Ayew who notched 11 goals in all competitions last season looks like he will be one of the players who will definitely be leaving the club.
Ayew will be looking to extend his stay in the top-flight and he will add quality to the Fulham side as he has vast experience in the Premier League.
If Fulham manage to land this deal, they will also be acquiring a proven Premier League goal scorer for a relatively low price.
Ayew arrived in England when he joined Aston Villa from Ligue 1 outfit Lorient in 2015.
After being unable to impress with the Villains, he ended up moving to Swansea in January 2017 in a deal which saw defender Neil Taylor moving the other way.
Ayew could also bring in a winning mentality to the Cottagers as he has previously won the Ligue 1 title which he did with Marseille in 2010.