The 20-year-old centre-back has been linked with a move to the Premier League this summer, prompting the Portuguese club to offer the player a new contract.
Reds boss Brendan Rodgers is keen to bring Ilori to Anfield, and hoped that an offer in the region of £3million would be enough to convince cash-strapped Sporting to sell.
However, despite their growing financial problems the Primeira Liga outfit are determined to keep the player from Liverpool's clutches, and have knocked back an additional offer to take Oussama Assaidi on loan at Estadio Jose Alvalade.
As it stands, an agreement is far from being reached with Sporting Lisbon set to demand double the amount of money currently offered by Liverpool, after talks with a delegation in Portugal over the weekend reached an impasse.
"The doors are open to negotiate, but for a just price," a source close to the Portuguese club confirmed to Sky Sports. "We are not disposed to sell any of our players this summer."
Jamie Carragher's retirement at the end of the 2012/13 campaign leaves Liverpool short on central defensive options ahead of the new season, with some question marks still surrounding the long-term futures of Martin Skrtel and Daniel Agger.
Rodgers has been linked with a move for Swansea's Ashley Williams, but might now consider an improved bid to lure Ilori to Merseyside.