Italy target Cesare Prandelli
World Cup winner Lippi has long hinted he will step down after the finals, which start on June 11, and Prandelli has been front-runner with the bookmakers for months after guiding the Florence club to the Champions League last 16 this season.
"Given that the request regards the national team, Fiorentina director Andrea Della Valle has told (Italian football) federation president Giancarlo Abete that the decision lies with the coach," a Fiorentina statement said.
"In the interests of the team and the fans, Fiorentina ask that the situation is cleared up as soon as possible so we can plan for the future."
Prandelli, who earlier this month said he was committed to his club, had no comment on Thursday when he prepared to fly with Fiore to Canada for a post-season tour of friendlies.
Amid confusion over Lippi's strategy, Abete has consistently said a decision on who would coach the Azzurri in the Euro 2012 qualifiers would be made before the team flies to the month-long World Cup on June 8.
Lippi, who quit after Italy's World Cup triumph in 2006 before returning two years later, has never explicitly said he would again be leaving but Abete's contact with Fiorentina has revealed the current coach's plans.
Prandelli, Serie A's longest-serving coach having been at Fiorentina since 2005, also notably managed Parma and AS Roma for a brief spell before leaving to care for his sick wife who later died.
Abete had said any replacement for Lippi would be Italian but his options were limited given Chelsea's Carlo Ancelotti and Manchester City's Roberto Mancini, who had both previously expressed interest in the job, have only just completed their first seasons with their clubs.
That left Prandelli as clear favourite despite Fiorentina enduring a poor campaign in Serie A where they finished 11th.
The fans blamed the team's slump on their good European run, a drugs ban for striker Adrian Mutu and tensions between Prandelli and the club's board.
The 52-year-old Prandelli, famed for his slicked-back hair and purple puffer jacket, had been seen in the media as a candidate for the Juventus job before the faltering Turin side chose former Sampdoria boss Luigi Del Neri on Wednesday.