Leicester were expected to sign the Algerian, but the Foxes offer of £25m was £9m of the Portuguese club's valuation.
Slimani - an international team-mate of Riyad Mahrez - scored 27 goals for Sporting last season and has also been linked with West Brom.
Leciester are also interested in signing Sporting's Adrien Silva and are hoping to seal a deal for the player before the transfer window closes.
The player yesterday 'confirmed' his move to the King Power Stadium, but it is not known at what stage the deal is at.