The Daily Express reports the Scot is one of the managers on the Londoners’ radar. Clarke held talks with Fulham in 2015 but decided to remain at Reading.
Despite the Cottagers ownership changing, there are still “influential figures” who rate the 55-year-old, according to the Express.
Clarke has won plaudits for his work with Kilmarnock, taking them from the Scottish Premiership basement to the upper echelons of the top flight and challenging for Europe.
His wealth of experience south of the Border - coaching spells with Chelsea, Liverpool, Newcastle and West Ham as well as management gigs with Reading and West Brom - makes him a strong candidate.
Clarke this week reiterated his desire to return to England with his comments coming in the wake of a row over sectarian abuse aimed at the Kilmarnock boss during his side’s recent William Hill Scottish Cup clash with Rangers at Ibrox.
The Rugby Park side would not want to lose Clarke, but would be well compensated due to the ex-St Mirren defender signing a new contract recently.
Fulham currently sit ten points adrift at the bottom of the English Premier League. Slavisa Jokanovic was tasked with leading the club into the top flight but despite spending more than £100 million on players he was sacked in November with the club bottom of the table. His replacement Ranieri managed just three wins and three draws in 17 games.