The Sun report that Kelly forged a close relationship with the new Liverpool boss during Hodgson's three-year stay at Craven Cottage.
And the former England goalkeeping coach played a major role behind the scenes in helping Fulham reach the Europa League Final last season.
Kelly brought out the best in Mark Schwarzer, who is currently being linked with a move to Arsenal.
Hodgson would love Kelly to have the same impact with Pepe Reina and bring through the other young keepers at Anfield.