The Daily Express says Fulham will attempt to lure the Ivory Coast manager to Craven Cottage if Hodgson decides to leave by promising a bumper transfer kitty funded by last month's sale of Harrods.
Chairman Mohamed Al Fayed banked £1.5b from the sale of his flagship Knightsbridge store in early May.
But he is understood to be prepared to provide significant funds to lure Eriksson.
Hodgson is hot favourite to land the vacant position as Liverpool manager. But he is also leading the candidates to succeed Fabio Capello as England boss should he be relieved of his post.
Understandably, he is waiting until after the World Cup to make his decision.