The Dutchman left United earlier this month after a short spell as interim boss and previously serving as assistant to Erik ten Hag.
Leicester sacked Steve Cooper on Sunday after just 12 Premier League games in charge, with the Foxes 16th in the table and only one point above the relegation zone.
"I'm proud, I'm excited. Everybody that I speak to about Leicester City Football Club is enthusiastic," said Van Nistelrooy in a club statement.
"The recent history of the club is impressive. I'm excited to start and to get to know everyone and give everything I can for the football club."
Leicester shocked the world to win the Premier League in 2015/16 and lifted the FA Cup for the first time five years later.
But they were relegated from the top tier two years ago before bouncing back at the first attempt under now Chelsea boss Enzo Maresca.
Two of Van Nistelrooy's three wins in four games as United's caretaker manager came against Leicester in the League Cup and Premier League.
The 48-year-old has previous managerial experience from one year in charge of PSV Eindhoven in 2022/23.
He will not begin his role until Sunday, leaving a caretaker team led by Ben Dawson to take charge of Saturday's visit to Brentford, but Van Nistelrooy will attend the game.
A prolific goalscorer, Van Nistelrooy enjoyed a glorious playing career, most notably at PSV, Manchester United and Real Madrid.
Leicester chairman Aiyawatt Srivaddhanaprabha said: "Ruud's experience, knowledge, and winning mentality will undoubtedly bring great value to us, and we look forward to supporting him in achieving success for our fans and our club."
© 2024 AFP