Argentinian Martino, 50, led Newell's Old Boys to the Argentinian Clausura title last season and recently severed his ties with the club.
Martino is from Rosario, where Barca's World Player of the Year, Lionel Messi, was born, and was the favourite player of Messi's father during his days as an attacking midfielder with Newell's Old Boys.
He also played for Spanish side Tenerife as well as Lanus in his homeland, Barcelona SC in Ecuador and O'Higgins in Chile.
Martino, who is nicknamed Tata, was manager of Paraguay from 2007 to 2011 - guiding them to the quarter-finals of the 2010 World Cup - before returning to Rosario-based Newell's Old Boys as boss for the last two seasons.
Vilanova, 44, stepped down last week to continue treatment for throat cancer after leading Barca to the La Liga title in his first season since taking over from Pep Guardiola.
A host of coaches had been linked to the Barcelona job, 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.