Martino, 50, who recently led Newell's Old Boys to the "Final" title in his native Argentina, and the 43-year-old Enrique, a former Spain and Barca player who joined Celta Vigo last month, have emerged in local media as the front runners for the coveted post.
Vilanova, 44, was forced to step down last week as his treatment for throat cancer means he will be unable to perform his duties.
Puyol told a post-training news conference he was sure sporting director Andoni Zubizarreta and the club's board would hire a coach able to adapt to Barca's possession-based ethic.
"We know about his career, he is a very good coach," the shaggy-haired centre back said of Martino.
"He likes Barcelona's playing style," he said, adding that he remembered Spain's quarter-final against Martino's Paraguay at the 2010 World Cup, which they edged 1-0, as "a very difficult, very hard game".
On Enrique, who may be a more controversial choice given that he also had a stint playing for Barca's arch rivals Real Madrid, Puyol said: "Everyone knows that I have a very good relationship with Luis.
"I think he has what it takes but that's the same for the other candidates who have been mentioned.
"I don't know who will come ... we are ready to help whoever comes in any way we can."
Barca need to move swiftly to appoint a successor as they launch their bid for a fifth La Liga title in six years in less than a month.
One reason Martino is seen as a likely candidate is that he is from the city of Rosario where Barca's World Player of the Year Lionel Messi was born.
Messi started out in the youth ranks at Newell's where Martino was also a player.
Enrique's chances, meanwhile, may also be hampered by the fact that he only recently agreed to join Celta as well as memories of an unsuccessful stint at Serie A side AS Roma during the 2011-12 campaign.
Puyol was also asked about his recovery from injury after he was forced to miss most of last season due to various ailments.
"I am getting better," the 35-year-old said.
"I would like to get back to training with my team mates again but I cannot just yet.
"I am not setting a date to return. I just want to get fit. It has been a very tough year with three operations and what I don't want to do now is rush things.
"It's about getting properly fit and when I return I want to be at 100 percent."
Barca have said they want to sign another centre back before the transfer window closes at the end of the next month and that Paris St Germain's Brazil captain Thiago Silva is their first choice.
"Thiago Silva is one of the best in the world and I think he would adapt well here," Puyol said.
Barca cancelled a pre-season friendly against Polish side Lechia Gdansk at the weekend because of Friday's announcement that Vilanova was leaving.
They are due to play German and European champions Bayern Munich, who are managed by former Barca player and coach Pep Guardiola, in a friendly in the Bavarian capital on Wednesday.
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