The Scotland captain was handed his debut by Berti Vogts and has since played under Walter Smith, Alex McLeish, George Burley and Craig Levein, with Billy Stark taking caretaker charge of tonight's friendly.
Fletcher said: "If it means taking a bit more time to get the right choice, then take as much time as they need.
"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: "We are all disappointed Craig left, but we are working with an 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."