According to widespread media reports in Spain on Monday evening, the Argentinian is set to be confirmed as the next Barca boss on a three-year contract following talks.
The 50-year-old led Newell's Old Boys to the Argentinian Clausura title last season but recently severed ties with the club. Barcelona have made no official announcement.
The Spanish champions are searching for a new coach after announcing on Friday night that Vilanova was stepping down from the post as he continues his battle with cancer.
The 44-year-old, who led Barca to the Primera Division title in his only season in charge after replacing Pep Guardiola last summer, had a tumour removed from his throat in late 2011, but suffered a relapse a year after.
He underwent another operation on his parotid gland in December before beginning a course of chemotherapy and radiotherapy in New York.
A host of coaches had been linked to the vacant position, including former Barcelona B boss Luis Enrique, Andre Villas-Boas, Marcelo Bielsa, Michael Laudrup, Rubi and most recently Guus Hiddink, who on Monday resigned as coach of Anzhi Makhachkala.
However, it seems Martino is now poised to take the reigns at the Catalan giants.
Barca captain Carles Puyol was asked about the possible options at a press conference on Monday morning and he stressed the importance of bringing in a coach who believes in the club's "philosophy".
He said: "I think all the names that have been mentioned are capable of coaching this team. We all know that Zubi (sporting director Andoni Zubizarreta) and the board are working hard and we're sure they'll bring us the best coach.
"We have a certain way of playing; a philosophy. The coach that comes in will follow that line and I'm sure that Zubi and the directors are looking for a coach that can adapt to our characteristics.
"I know (Martino) is a very good coach, who likes the style of the club. And you all know that I have a very good relationship with Luis (Enrique) - he's one of the most accomplished names - but I like all the others who have been mentioned."
Puyol, 35, also paid tribute to Vilanova, who inspired Barcelona to a 22nd Primera Division title despite missing a number of matches in the second half of the campaign as he underwent treatment in the United States.
"It's really tough news, not just because it's happened to a coach or a colleague, but because it's an illness," Puyol added.
"The person is the most important thing. We remember how he talked to us, how he asked us to give everything, asked us to fight. We certainly did that and we did it even more because of him."