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The move is subject to Premier League and international clearance, with Leicester reported to have an option to buy Under when his loan expires.
The 23-year-old is due to join up with his new team-mates and manager Brendan Rodgers ahead of Sunday's Premier League game against Burnley.
"I can't wait to go to Leicester and start training. I've always wanted to play in England and this is a great opportunity to play in the Premier League," Under told Leicester's website.
Under made his debut for Turkey aged 19 and will now play alongside his international team-mate Caglar Soyuncu.
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"I will try to help my team-mates on the pitch and off the pitch and I believe that I can improve my performance every day," Under said.
"I feel very good because my best friend plays for Leicester City. I am very happy to be able to play with Caglar and I always watched Leicester's games because of him."
Under began his senior career with Altinordu before moving to Istanbul Basaksehir in 2016 and then making the switch to Italy a year later.
He has made 70 league appearances for Roma, scoring 13 goals.
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