The 24-year-old winger will see his current deal at Camp Nou expire in June and he is now free to sign pre-contract agreements with foreign clubs.
Dembele representatives are believed to have held talks with Barcelona's hierarchy for several months, but as things stand, the Frenchman is set to become a free agent next summer.
Speaking to Sport, Alemany said: "[Dembele] knows very well what the club thinks. He has our offer on the table and he knows that we should resolve this situation quickly.
"We hope that it can be for the good and he stays with us because he is an important player and there will be updates soon."
Dembele's agent, Moussa Sissoko, has claimed that the winger's contract standoff with Barca is not due to financial reasons, and has instead suggested that Xavi's management has made discussions difficult.
The 2018 World Cup winner played the full 120 minutes in Barcelona's 3-2 extra-time defeat in the Spanish Super Cup semi-final against rivals Real Madrid on Wednesday. body check tags ::