The Argentinian is under pressure after the Catalan club slumped to a 3-1 defeat against Real Sociedad last weekend, which allowed rivals Real Madrid to take their place at the top of La Liga.
Reports have also surfaced in Spain that Martino, who replaced Tito Vilanoa as Barcelona boss in the summer, has lost the dressing room.
Now Sport claim that the club are drawing up a list of potential successors should Martino get the chop in the summer, with former Chelsea and Tottenham boss Andre Villas-Boas amongst those being targeted.
The Portuguese boss joined Tottenham in 2012 after his time in charge of Chelsea was cut short - but he failed to see out his second season at White Hart Lane and left by 'mutual consent' in December.
Villas-Boas has previously credited former Barcelona boss Pep Guardiola as the main influence on his career, and worked under Sir Bobby Robson, who took charge at the Nou Camp between 1996 and 1997.
Also on the list is former midfielder Luis Enrique and and Athletic Bilbao boss Ernesto Valverde.
Those three managers are seen as 'continuity options' who can adhere to the club's passing philosophy.
Barcelona have also included two managers they believe will come in and change the way they play - Borussia Dortmund boss Jurgen Klopp and Atletico Madrid's Diego Simeone.