Fulham showed they hardly needed last season's top scorer when they romped to a 5-0 home win over Norwich City on the Premier League's opening day but Jol's post-match news conference was still dominated by Dempsey's no-show.
"If you want to move you don't want to take risks, you don't want to play because you could get an injury. That's a very awkward situation for us and for me and for him as well," Jol told reporters.
"Maybe Clint knows more than us but we feel and the players feel and everyone else feels that he should stay here if there is no other bid because there is no other bid.
"In a normal world you have to stay where you are because you signed your contract and hopefully he will play for us again." added Jol.
Dempsey scored 17 league goals for Fulham last term despite not being an out-and-out striker and United States boss Jurgen Klinsmann said in May he hoped the 29-year-old would move to a Champions League side after his fine campaign.
Liverpool, who are not in the Champions League this season and started life under new manager Brendan Rodgers with a 3-0 defeat at West Bromwich Albion on Saturday, have long been linked in the media with a bid for Dempsey along with Arsenal.
Arsene Wenger's Arsenal side, who drew 0-0 with Sunderland, are in Europe's elite competition and need fresh forwards after selling leading striker Robin van Persie to Manchester United.
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