The 23-year-old midfielder becomes Norwich's first signing of the January transfer window.
The deal will go through officially once the paperwork has been approved "by the relevant footballing authorities", the Canaries confirmed.
Leeds confirmed last week they had "reluctantly" accepted an offer for Howson, whose deal at Elland Road expired in the summer.
Howson made his Leeds United debut against Hull in 2006, before becoming the club's youngest captain since Billy Bremner when he was given the armband as a 19-year-old in April 2009.
Norwich manager Paul Lambert said: "I think he has been brilliant for Leeds - how he has played but also being their captain brings its own demands. It's a fantastic club Leeds.
"I think he can add goals to our midfield which I think we need and he can do that. He'll get better as a player the higher the level he goes up. If he does that and everything goes well, he'll be a great player for us."
Howson will wear shirt number 24 for Norwich. He has been out of action since December because of a knee injury suffered playing for Leeds against Millwall, but is hopeful of being fit soon.
The England Under-21 international had been mulling over signing a new deal with Leeds, but opted not to as they could not guarantee him top-flight football.
"The reason is Premier League," he told LUTV.
"I am going to a very good club with good support, but at the same time I know I'm leaving a club I've loved from being a young boy. That's not going to change.
"Everyone at Leeds is wanting Premier League football. The ambition is always there, but that is not guaranteed."
The departure of Howson, though, has proved to be a tipping point for many fans, who used Saturday's 3-1 win over Ipswich as a chance to protest against chairman Ken Bates and a perceived lack of ambition and investment.
"It makes it harder when you see a reaction like that. You realise you mean something to them," Howson added.
"I was the club captain and from the area, but it was me that was still wanting to play in the Premier League and, from the club's point of view, I would have done the same thing if I were in their shoes."
