Crewe chairman John Bowler has said he will be looking for an experienced manager to replace Steve Holland, who was relieved of his duties on Tuesday.
Former Alex boss Dario Gradi has been placed in caretaker charge, but does not want to take the job permanently.
Bowler told BBC Radio Stoke: "This isn't a role for someone to come in and learn on the job.
"We want to bring in an experienced, proven manager who wants to work in this environment."
Gradi relinquished control of the Alex first-team in the summer of 2007 when he took up a new role as the club's technical director.
His 23-year reign at the Alexandra Stadium was at the time, the longest of any manager in the Football League.
"I am looking forward to the excitement," said Gradi.
"It's a difficult job and one that I don't want to do for too long. I missed it when we played at Anfield in the League Cup, but I haven't missed it for the rest of the time," he added.
Gradi's role in charge of the first-team was taken by Holland, who narrowly managed to avoid relegation on the final day of last season.
Holland was relieved of his duties after a poor start to this season, which has seen the Alex slump to the bottom of League One with just nine points from 16 games.
"Steve is a guy who we have enormous respect for and his contribution to this football club is second only to Dario," said Bowler.
"Had he had the rub of the green we may have been further up the league and relieving him of his duties was the hardest decision this board has ever had to take.
"We value him and want to talk to him about another job within the club, but that has to be his decision."
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