Avram Grant in frame to succeed Paul Hart at Portsmouth
Avram Grant was appointed director of football in October
Former Chelsea manager Avram Grant must wait before he learns whether he will be appointed as Portsmouth's next boss.
Executive director Mark Jacob said the club would not be rushed into appointing a replacement for Hart.
The 54-year-old Israeli is already at the club as director of football, with his work permit expected to be confirmed by Friday.
Hart, who left the struggling club on Tuesday, is considering an offer to become the club's technical director.
Portsmouth, whose owner Ali Al Faraj is set to arrive in the UK by the end of the week, issued a statement on Tuesday announcing Hart had declined the role to become responsible for players aged 18-21.
But a club spokesman said on Wednesday: "He had a further conversation with (chief executive) Peter Storrie and said he would go away and get back to us.
"Hopefully he will stay at the club and work in the capacity of technical director. Everyone at the club would be delighted if he stayed."
Grant and former Peterborough boss Darren Ferguson have been installed as the favourites to succeed Hart, with the south coast club hosting Manchester United, managed by Darren's father Sir Alex Ferguson, on Saturday.
But Kevin Ryan, secretary of Portsmouth Supporters' Club (central branch), said the fans would prefer to see someone with top flight experience at the helm, like Grant or former Charlton and West Ham boss Alan Curbishley.
"I think managing Portsmouth is a job for a very experienced manager," Ryan told BBC Sport.
"I don't think it's a job for a young manager at the moment. Darren Ferguson seems like he has potential but I don't think Portsmouth is the right time for him.
"I think it has to be Curbishley or Avram Grant. I would have liked to have seen Gordon Strachan (now Middlesbrough manager) in the job.
"A month ago Strachan said he was ready to go back into football and that's when Portsmouth should have moved. Strachan has the commitment and passion that Portsmouth fans love. We love to see someone jumping around the touchline."
I don't think Hart should have been given the job in the first place. He did a great job last season, but he wasn't the right man to take us forward
Kevin Ryan, secretary of Portsmouth Supporters' Club
Ryan said the fans sympathised with Hart because of the "difficult job" he undertook when he succeeded Tony Adams in February 2009.
He kept Portsmouth in the top flight last season and was appointed on a permanent basis in July.
But Portsmouth, currently bottom of the Premier League, lost several high-profile players over the summer and were hit with a transfer embargo for debts owed to other clubs.
"I don't think the players are good enough, but I think Hart should have gone a month ago," added Ryan of the man who guided the club to just two Premier League victories this season.
"I don't think he should have been given the job in the first place. He did a great job last season, but he wasn't the right man to take us forward."
Former Portsmouth manager Harry Redknapp has already given his backing to Grant.
Grant is currently awaiting his work permit so he can take up his position as director of football, a role he previously held at Fratton Park under Redknapp
The former Chelsea boss, who succeeded Jose Mourinho and guided the Blues to the Champions League final in 2008, returned to Portsmouth in October.
Portsmouth have three consecutive home matches ahead, with Manchester United's visit on Saturday followed by a Carling Cup clash with Aston Villa on Tuesday and a league tussle with Burnley four days later.
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