The Anfield club had a €3m bid rejected by the Portuguese giants earlier this summer, but the report claims Rodgers is set to make a final effort to sign the player.
The Liverpool manager has been forced to go back into the market for defensive cover following the injury to Sebastian Coates.
The Uruguayan centre-back could miss the entire season with a knee injury according to Liverpool.
That leaves Martin Skrtel - himself the subject of transfer speculation - Daniel Agger and Kolo Toure as Liverpool's recognised defensive options.
Ilori has been a target for Rodgers for much of the summer, but the report claims the Liverpool boss would have to more than double his initial offer to convince Sporting to part with the youngster.
The 20-year-old has only made a handful of league appearances for Sporting, but impressed for the Portugal under-20s in international competition.
Barcelona have also been mentioned as possible suitors for Ilori, who is valued at closer to €7m by his club.