Lerner has launched the process of drawing up a list of potential candidates after O'Neill ended his four-year reign.
Sources close to the American multi-millionaire indicate he would like any successor in place for Villa's second Premier League clash of the campaign at Newcastle on August 22.
Reserve coach Kevin MacDonald will be in charge for the league opener at home to West Ham on Saturday.
The former Liverpool midfielder could also be at the helm for next Thursday's Europa League tie at Rapid Vienna if the new manager is still not in place.
Former England coach Sven-Goran Eriksson has made it known he would be keen to step into the Villa Park hotseat.
The Swede is known to be eager to return to work after his spell as Ivory Coast coach during the World Cup.
Four years ago Eriksson indicated he would be ready to take over at Villa when the victim of a 'fake sheikh' sting by a Sunday tabloid.
A source close to the former Manchester City boss said: "Of course Sven would be interested in the Villa job. He is keen to return to the Premier League."
Eriksson's agent Athole Still said: "There has been no contact from Aston Villa. Sven gets associated with a lot of big jobs."
Lerner will not be short of interested parties despite the apparent budget restraints which led to O'Neill deciding to quit the midlands club.
United States coach Bob Bradley, Ajax coach Martin Jol and Germany legend Jurgen Klinsmann are all featuring highly in the bookmakers' odds to replace O'Neill.
Villa's chief executive Paul Faulkner met with the club's players not on international duty at Bodymoor Heath on Tuesday.
He explained the events of the last 24 hours and the immediate plans for the future.
But midfielder James Milner - a key figure in O'Neill's decision to step down - was not present as he is part of the England squad preparing for the game with Hungary at Wembley.
It was a failure to gain assurances from Lerner that all the money from Milner's expected move to Manchester City would be ploughed into new signings which led to O'Neill saying enough is enough.
Lerner had previously backed O'Neill in the transfer market but this summer switched to a "sell-to-buy" policy.
There has also been speculation that O'Neill was concerned Ashley Young - also on England duty - was about to be sold to Tottenham and keeper Brad Friedel to Fulham.
But this was not at the forefront of O'Neill's decision to walk away from Villa, whom he guided to three successive top-six finishes.
Lerner's predecessor Doug Ellis, who appointed O'Neill as manager, is confident the current regime will find an ideal replacement.
Ellis said: "I'm saddened and most supporters will be saddened (at O'Neill leaving).
"But the club is bigger than any one man and, in Randy Lerner, Aston Villa have a good chairman.
"I know he will find a successor and I hope all supporters will back him and Paul Faulkner in finding the best possible man for the job."
Central defender Curtis Davies has hit out at O'Neill, claiming he was not given a fair chance to prove himself.
Davies, one of six players made available for transfer, said: "A lot of players have been frustrated by not having a fair crack of the whip and I'm one of them.
"There is no guarantee whoever comes in is going to play you or not play you.
"But if you feel you have got a chance, then you are happy. As long as someone who comes in who is fair, I am sure people will be happy.
"I felt I didn't have a chance."
Davies added: "It is not an ideal situation four days before the season, not to have a manager but you've got to deal with what you've got.
"You've got good players at this club and Kevin MacDonald is a good coach.
"Hopefully he has got enough time to put his stamp on us and we can pull together for him."