Napoli chairman Aurelio De Laurentiis hit out at coach Maurizio Sarri for failing to clarify his future, saying: "Time has run out."
Sarri's side beat Crotone 2-1 in their final Serie A game of a season in which they finished second after having topped the table earlier in the campaign.
Such an impressive campaign has led Chelsea to consider Sarri as a possible replacement for Antonio Conte, who is still expected to depart Stamford Bridge despite the Blues ending the season by winning the FA Cup on Saturday.
It was understood that Sarri would be remaining with the Serie A outfit, however, following positive talks ahead of the Crotone clash, but De Laurentiis has now suggested that might not be case.
When asked whether Sarri would remain at the club, De Laurentiis told Kiss Kiss Radio: "Ask him. He has always sent me back to his agent [Alessandro] Pellegrini, who never gave me an answer.
"Time has run out, there is no date. We have dealt with this problem anyway, we have to go on without fear and without 'if'."
"I have to take responsibility," he added. "I'm not saying he'll leave. There is a release clause, we'll see what will happen."
Sarri's post-match comments had an air of finality about them, but he stopped short of giving a clear indication about where he will be next season.
"In life, everything comes to an end," he told reporters. "Sometimes it's better to finish stories while they are good.
"When I arrived, the team had finished 24 points off the top. Now, we are four points behind.
"We have not reached the target, but the journey has been fantastic. Tomorrow I will talk to my family before anyone else."
"The [club] president needs a response quickly," he added, before hinting that Chelsea could be among his preferred destinations by saying: "I don't think I could go to another Italian team immediately.
"The memory of Napoli would be too strong."