According to the Sunday Mirror, the Cottagers are still hopeful Hodgson will remain with them next season, after guiding them to mid-table security in the top-flight and to the final of the Europa League.
However, should the highly respected coach choose to move to Anfield, Fulham's top-brass are reportedly already making plans for his replacement.
Former England, Mexico and Manchester City coach Eriksson is currently in charge of the Ivory Coast, although his contract expires at the end of the World Cup and it is believed he would relish another opportunity to manage in the Premier League.
Following their successful run in the Europa League last season, interest in Hodgson has intensified, and Liverpool have made him their number one target to replace Rafael Benitez, who left to join Inter Milan earlier this month.
However, it is also reported that Hodgson has been included on a shortlist for the next England manager as Fabio Capello looks set to leave after the tournament in South Africa.