"The club has been disappointed with this season's results, performances and the general message these have sent to our fans," it said.
Villa chairman Randy Lernersaid: "We need to be clear and candid with ourselves and with supporters about what we have lacked in recent years.
"The most immediate action that we can take is to look carefully at our options in terms of bringing in a new manager who sees the club's potential and embraces our collective expectations."
The statement came after a meeting between McLeish and Lerner, who flew in from the United States with chief executive Paul Faulkner for the showdown with the Scottish manager.
Faulkner said: "We'd like to thank Alex for his hard work and efforts throughout this tough season.
"We are mindful of the club's need to prepare thoroughly for the new campaign and for the new manager to be able to begin working with the board so that he is quickly immersed in these preparations and also in the long-term objectives of the club."
Disillusioned Villa fans turned their fury on beleaguered McLeish throughout the club's 2-0 last-day defeat against Norwich at Carrow Road.
McLeish had insisted he would not quit his post , according to the Birmingtham Mail, despite the claret and blues finishing with less points than local rivals Birmingham City did when McLeish led them to relegation a year ago before walking out to join Villa.
Villa finished 16th, two places and two points above the relegation zone and with their final 38 points tally one less than the Blues' total of a year ago.
Norwich City manager Paul Lambertis the favourite to take over at Villa Park, according to the Eastern Daily Press, citing odds of 6 to 4 at Skybet and 7 to 4 at Bet Victor.
Wigan boss Roberto Martinez is second favourite, the Norfolk daily added.
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