West Brom have told Bakary Sako to lower his wage demands if he wants to move to The Hawthorns.
Highly-rated Atletico Madrid midfielder Saul Niguez has signed a new five-year contract.
A brief club statement issued on Saturday afternoon read: "The board of Aston Villa are delighted to confirm that Paul Lambert has been appointed Villa manager."
Lambert led Norwich to a 12th-place finish in the Premier League last month but angered the club this week when he offered his resignation following interest from Villa.
The Canaries refused to accept the Scot's resignation during the week but admitted on Friday their search for a new boss had begun, with Lambert wanting to leave.
Norwich released a statement on Saturday that read: "We would once again like to place on record our sincere gratitude to Paul for everything he has done for the club - everybody at Carrow Road wishes him all the best for the future.
"Some media reports are suggesting we have agreed compensation for Paul's services with Aston Villa, however this is not the case. We will continue to discuss this matter privately with Aston Villa.
"Meanwhile the search for the next Norwich City manager is already actively underway and the club has been inundated with applications for and interest in the job of leading this great club forward in the Barclays Premier League.
"The next Norwich City manager will have a great squad of players, a robust and competitive budget and excellent facilities to work with - as well as the superb backing home and away of our fantastic supporters."
Villa finished the season in 16th place, sacking manager Alex McLeish just days later.
Lambert's appointment will only increase speculation that Norwich striker Grant Holt, who recently handed in a transfer request, will also be heading to the West Midlands.
It is believed Lambert will be given money to spend after McLeish spent last season reducing what he described as an "astronomical'' wage bill.
Lambert will also be able to use the money he generates from the sale of current Villa players.
Lambert will be charged by club owner Randy Lerner with the task of revitalising Villa after two seasons in the doldrums.
After three successive top-six finishes under Martin O'Neill, Villa have dropped too close to the Barclays Premier League bottom three for comfort.
Gerard Houllier replaced O'Neill but had to quit last summer on health grounds, while former Birmingham boss McLeish was never accepted by the Villa fans.
Lambert fits the profile of being a relatively young, hungry and ambitious manager after working his way through the lower leagues before transforming Norwich's fortunes.
That is certainly the feeling of Villa midfielder Barry Bannan, who said before Lambert's appointment: "He has done very well with Norwich and the clubs he was at before then.
"He has a good record up until now so hopefully if it is Lambert he can be as successful with us as he has with his previous teams.
"Obviously last season wasn't the best for a team as big as us. We should have been way higher up the league and I think (the board) see Lambert as an up-and-coming manager that is doing really well, so it is going to be a massive lift for everyone, the fans and the players as well.
"It is a fresh start for everyone again and hopefully in pre-season everyone is going to have a chance to prove their worth to the club.''
Leading German manager Ottmar Hitzfeld - now manager of Switzerland - has praised the qualities of Lambert, whom he managed when in charge of Borussia Dortmund.
Hitzfeld, who led Dortmund and Bayern Munich to Champions League success, said: "I have always followed Paul since he left Dortmund, and I congratulated him on his promotion to the Premier League with Norwich City.
"As a player, he always thought like a coach. He was a leader so it is no surprise that he has become a successful manager.
"Paul was a very good team player. That is one of the most important qualities as a coach and manager of a team."
Lerner also held talks with former Manchester United star Ole Gunnar Solskjaer, who opted to stay at Molde in Norway for family reasons, although he was not formally offered the Villa post.
Villa also spoke with advisors of Wigan boss Roberto Martinez, who rejected the chance to replace Houllier 12 months ago.
But Latics boss Dave Whelan claimed Martinez, who held talks with Liverpool, was staying put.
Norwich's hunt for a new boss could see them turn to Cardiff's Malkay Mackay - the former Canaries defender is the early favourite for the job with the bookmakers.
Birmingham manager Chris Hughton has also already been linked with the post, while another former Canaries star, Steve Bruce, is currently out of work.
Norwich chief executive David McNally confirmed the search was under way even before Saturday's announcement.
"We are looking for a new manager, that is likely what we concentrate on in the next few weeks,'' McNally told Sky Sports News on Friday.
"We were in a situation where we were approached by another club to speak to Paul Lambert, we didn't grant them permission and we were very clear we wanted to keep our manager who has been incredibly successful.
"When it was indicated by our manager that he would like the opportunity to speak to this particular club the dynamics changed, and then it was about talking to them about compensation, agreeing compensation and sorting out a time frame.''
He added: "Paul Lambert has been fantastic for this football club for three years and he will quite rightly take his place in Norwich City history and we will always be very grateful to Paul for his achievements here.''
Paul Lambert factfile
1969: Born August 7 in Glasgow, Scotland
1980: Begins his career at Linwood Rangers Boys' Club as a youngster.
1985: Joins St Mirren and goes on to make 227 appearances for the Scottish Premier League side, scoring 14 goals in the process.
1993: Motherwell manager Tommy McLean signs Lambert and Motherwell go on to have several successful years, qualifying for the UEFA Cup two seasons running.
1996: Joins Bundesliga side Borussia Dortmund.
1997: May 28 - Wins the Champions League as Dortmund overcome Juventus 3-1 in the final.
November 7 - Returns to Scotland as he signs for Celtic for a fee in the region of £2million.
2005: Retires from playing and begins his managerial career at Livingston, where he also comes out of retirement to cover injury shortages.
2006: February 12 - Resigns as Livingston manager after winning just two league games.
June 30 - Appointed manager of Wycombe.
2008: May 20 - Resigns as Wycombe boss after the club are eliminated from the League Two play-offs.
October 9 - Becomes manager of League One Colchester after the departure of Geraint Williams.
2009: Leaves Colchester and replaces Bryan Gunn as Norwich manager, having opened the season with a 7-1 win over the Canaries.
2010: Leads Norwich to the League One title and promotion to the Championship before signing a new improved contract. Also named League One Manager of the Season.
2011: Norwich are promoted to the Premier League after finishing second in the Championship behind QPR. Lambert is also named Championship Manager of the Season after guiding the Canaries to the first back-to-back promotions from the third tier to the top flight since Manchester City in 2000.
2012: May - Leads Norwich to a 12th-placed finish in their first season in the Premier League.
June 2 - Named new manager of Aston Villa.