Jorge Messi met with club president Josep Maria Bartomeu on Wednesday to discuss his son's desire to leave Camp Nou.
The talks came after a week-long stalemate which began with Messi's bid to unilaterally rescind his Barca contract on Aug. 25, believing that a clause enabled him to leave for free.
Barcelona insist that clause expired in June and that Messi's contract -- which runs until 2021 -- is still valid. They say that if Messi wants to leave now, he or his next club must pay his €700 million release clause, or face legal action.
Club sources told ESPN there was a positive atmosphere at Wednesday's meeting, with both sides keen to find a resolution.
Barcelona believe that Messi's position isn't due to doubts over their sporting project, but rather his search for a new challenge, both on and off the pitch, at this stage of his career.
The club argue that Messi can seek that challenge, if he wishes, from 2021, either at Camp Nou, where a new president and board of directors will have been elected, or elsewhere.
Messi is yet to make any comment on the meeting, and is thought to be discussing the situation with his father before deciding on the next step.
Sources told ESPN that both parties have agreed to meet for further talks, although a day and time has not yet been set.
Earlier on Wednesday, father Jorge told reporters it would be "difficult" for his son to stay at Barcelona.
The 33-year-old has been refusing to attend training sessions with new Barca coach Ronald Koeman this week.
Manchester City are favourites to sign the six-time Ballon d'Or winner if he leaves, with sources telling ESPN the club have been working on a deal that would see Messi play for them before joining New York City FC.