Algerian international Slimani joined the former Premier League champions from Portuguese side Sporting in 2016 and scored 13 goals in 46 appearances in all competitions.
Slimani, 30, struggled to nail down a starting spot in the Leicester squad and was loaned to Newcastle United last season.
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Fenerbahce, who were runners-up to Galatasaray in the Turkish Super Lig last season, kicked off their campaign with a 2-1 win at home to Bursaspor on Saturday.
Leicester, who were beaten 2-1 by Manchester United in their season opener on Friday, host Wolverhampton Wanderers next weekend.
City's Leonardo Ulloa ended his four-year stay at the club to join Mexican outfit Pachuca.
Ulloa, who joined Leicester from Brighton and Hove Albion, scored 20 goals in 101 appearances for the club and helped them win the Premier League title in the 2015-16 campaign.
"Leonardo's contribution to the club has been a significant one and he will remain a popular figure among players, staff and supporters alike," Leicester said in a statement.
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