Fletcher, 28, was handed his debut by Berti Vogts and has also played under Walter Smith, Alex McLeish, George Burley and Levein with Billy Stark taking caretaker charge of Wednesday's friendly in Luxembourg.
Levein lasted almost three years in the job before paying the price for a poor competitive record of three wins from 12 matches.
Scotland's next World Cup qualifier is against Wales on March 22 and Fletcher would like a new man in place for the friendly against Estonia the previous month.
But SFA chief executive Stewart Regan has made no promises about making an appointment for the Pittodrie clash on February 6 and Fletcher feels the decision is too important to rush.
The Manchester United midfielder said: "If it could happen before Estonia then that would obviously be beneficial because a friendly game in preparation before a qualifier is always ideal.
"A new manager taking over as soon as possible is always ideal. But if it means taking a bit more time to get the right choice looking at it long term, then take as much time as they need.
"Maybe there are a lot of candidates and for whatever reason it takes time, the most important thing is getting the right man who can provide a bit of continuity and take Scotland forward not just for the next couple of years but hopefully five or six years and really stamp his authority and take the country places where we want to go."
Fletcher, who has 60 caps, added: "Managers have gone on because they have achieved success for Scotland, and there have been managers who have gone on because they have maybe not.
"We are all disappointed Craig left, I think all the players would have liked to have seen him stay and continue his work and hopefully it could have come through on the pitch, but ultimately that's not the case.
"Just now we are working with a really experienced man in Billy Stark, who has been fantastic in the few days we have been together. He is desperate to do well and desperate to get a win."
Fletcher has been impressed with Stark in recent days, adding: "Although I don't think he will be expecting to be named a new manager, he will be desperate to get a good result and who knows what could happen.
"He has done great things from the under-21s and got a lot of respect from all the players and even in the couple of days that he has been in charge, a lot of players have been very impressed by him.
"If Billy could get a good victory and there is not a manager appointed before the next game, and we get another good victory against Estonia, then he really throws his name into the hat.
"It would be a choice backed by me but I'm not here to pick managers, I'm here to play for my country whoever is in charge and give them 100% commitment whoever the SFA decide to pick."