Leicester said that Wague, who played in this month's African Nations Cup against Leicester's Ghanaian midfielder Daniel Amartey, was their second signing of the transfer window that closes on Tuesday.
The club have an option to buy the player, whose move is subject to international clearance, from his parent club Granada at the end of the season.
"I’m delighted to be at Leicester City and I can’t wait to get started. It’s a wonderful club and I’m excited to have the chance to show what I can do in the Premier League with the champions," the French-born player told the club website.
Wague, a former French youth international, has been on loan at Udinese since 2014, making 37 appearances for the Serie A club.
He also played alongside former Leicester player N'Golo Kante at Caen earlier in his career.
(Reporting by Alan Baldwin, editing by Toby Davis)
