Schwarzer, 42, played 12 times for Chelsea last season as backup to Petr Cech, but has only been an unused substitute four times this season.
"I want to publicly thank Mark Schwarzer because he's leaving us and he was a fantastic guy for us," said Mourinho.
"As a goalkeeper, when we needed him he was there for us, and as a person and a professional he was an amazing guy in our group.
"On behalf of myself, my staff and my players I want to say we are going to miss the big guy and we wish him all the best in his new life at Leicester."
Schwarzer, who had a brief spell in Germany in the mid-1990s, has been in England since 1996 and played for Bradford City, Middlesbrough and Fulham before moving to Chelsea in 2013.
He played 109 times for Australia and is likely to be involved when Leicester play Aston Villa in the Premier League next Saturday while first choice Kasper Schmeichel is injured.
Schmeichel's deputy Ben Hamer has been in Leicester's last six matches including their 1-0 FA Cup third round win over Newcastle United on Saturday.
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