The France international arrived at Camp Nou in 2017 immediately after Neymar was sold to Paris Saint-Germain, but has failed to sufficiently replace the Brazilian.
Injuries and confidence issues have blighted Dembele, yet new manager Xavi remains a key believer in the 24-year-old.
Dembele, who will be a free agent as of June 30, has 30 goals and 23 assists in his 126 appearances and has attracted the attention of Manchester United.
Reports of a swap deal with Anthony Martial have been mooted, with Spanish football expert Terry Gibson insisting his time could well be up in Catalonia.
"They are doing everything they can to keep him, but they have made it clear now this is their final offer," Gibson told Hawksbee and Jacobs.
"If he doesn't sign in this window, their new contract, he can go and they will sell him as quickly as they can.
"There was talk in the papers in Spain about a swap deal with Anthony Martial as well.
"At least they are trying to be active, they are trying to deal with the situation. Whereas in the past, they were just left with players nobody wanted or who didn't fit in with the club and nothing was happening.
"So at least they are trying to be active now."