The France international's contract with the Catalan giants expired at the end of June, with the two parties unable to come to an agreement over an extension.
There has been a host of speculation surrounding the World Cup winner's future, with Chelsea and Paris Saint-Germain both believed to be interested in signing him on a free transfer.
However, Dembele's first choice has allegedly always been to stay at Barcelona, and according to Italian journalist Fabrizio Romano, an agreement between the two parties is now close.
Romano claims that the attacker has 'accepted all clauses', with the new deal set to run until June 2024, and it will be on a reduced salary due to Barcelona's financial problems.
The journalist claims that Xavi has been pushing to keep hold of the former Rennes youngster, and an announcement on the new deal could arrive before the end of the week.
Dembele had allegedly wanted double the €6m (£5.2m) salary that had been placed on the table by Barcelona but has now accepted that will not be possible due to the team's financial issues.
The Catalan club's president Joan Laporta recently said that the club would not place a deadline on deciding the attacker's future.
"He is a person that we appreciate very much," Mundo Deportivo quotes Laporta as saying. "He has played at a very high level in the second half of the season and the coach is very satisfied with him.
"It is an economic question, he has answered us without saying 'yes' or a 'no'. We maintain a good relationship with the player and there is communication with the agents. If he accepts the conditions that we have presented, he will continue, if not, then no.
"We don't set deadlines for anything, not for signings, not for levers or anything. Now that we've activated the first lever, we've released quite a bit of pressure. We are working hard, both for players and levers."
The 25-year-old was initially told to leave Barcelona in January after refusing to sign a new deal, but he ultimately remained at the club and managed to work his way back into the side in the second half of the season.
Dembele enjoyed a productive 2021-22 campaign despite some injury troubles, scoring twice and providing 13 assists in 32 appearances in all competitions.
Meanwhile, he has scored 32 goals and registered 34 assists in 149 appearances for Barcelona since arriving from Borussia Dortmund in August 2017.
As well as Chelsea and PSG, Manchester United, Bayern Munich and Liverpool are believed to have shown an interest in Dembele over the last 12 months.
Xavi's side are also working on big-money deals for Bayern's Robert Lewandowski and Leeds United's Raphinha during the summer window, as the club look to challenge for major silverware next term after a difficult 2021-22 season.