A source told ESPN that Messi Sr. plans to travel on to Italy this week to deal with some "personal issues" and that no talks have yet been scheduled with Barca.
A meeting with new Barcelona president Joan Laporta has not yet been ruled out, but no date has been fixed to begin negotiations over a potential new deal for Messi.
Messi's current terms expire in June, when he turns 34, and he has said that he will wait until the end of the season before deciding on his future.
Laporta, who was elected for a second spell as Barca president on March 7, said after Saturday's 4-0 Copa del Rey final win over Athletic Bilbao in which Messi scored twice, that he was "convinced" the Argentine would stay put at Camp Nou.
The Barca president is currently putting the finishing touches to the contract offer, sources have told ESPN, having wanted to carry out an internal audit on the club's finances first.
More important than the wages on offer, though, will be Laporta's ability to convince Messi that he can compete for the Champions League if he stays in Catalonia.
However, with improved performances under coach Ronald Koeman bringing Copa success this weekend and helping Barca get back into the La Liga title race, there is renewed optimism he could stay.
The return of Laporta is also cause for hope. In contrast to his relationship with former president Josep Maria Bartomeu, Messi and his family are on very good terms with Laporta.
Messi Sr. took advantage of his visit to Barcelona to watch Barca B beat Villarreal B 3-2 at the Johan Cruyff stadium on Sunday in the Spanish third division.
He was sat with one of his other sons, Rodrigo, who represents United Stated international Konrad de la Fuente, one of the three scorers in the B team's win.
De la Fuente, 19, made his first-team debut earlier this season but sources have told ESPN that Barca will listen to offers for him at the end of the season, with several clubs in Spain and Germany interested.