The hosts, boasting half a dozen of the world's top forwards and the pre-tournament favourites, have struggled and have found the net only once in two draws against Bolivia and Colombia.
The Costa Ricans, with a young side under wily Argentine coach Ricardo La Volpe, shocked Bolivia 2-0 in Jujuy on Thursday to take second place in Group A behind Colombia and above Argentina.
Messi, his team-mates and coach Sergio Batista have come under fire from media and fans with former captain and 2010 World Cup coach Diego Maradona stepping in to demand more support for Argentina.
"We Argentines can't behave like this with the best [player] in the world," Maradona said, criticising reports attacking Messi.
An emotional Maradona, whose mother is in hospital, also told the sports daily Ole in an exclusive interview published on Sunday, that he would have struggled playing with three holding midfielders.
ONE GOAL
Batista plays Javier Mascherano, Ever Banega and Esteban Cambiasso in the middle of a 4-3-3 formation. Sergio Aguero, as a substitute against Bolivia, has scored their only goal of the tournament so far.
"[Javier] Pastore would be a good associate for Lio," said Maradona referring to the Palermo playmaker who shone this season in Serie A and to a lack of creativity in Batista's midfield. Pastore has yet to make even a substitute's appearance in the tournament.
Costa Rica, with a mainly under-23 team and only half the squad that reached the quarter-finals of last month's Concacaf Gold Cup in the United States, could upset the apple cart if Argentina do not improve.
La Volpe, whose side were invited to the tournament at the last minute after Japan pulled out, said their preparations included a psychological aspect.
"I'm focusing on how to counter a strong team who want to show the Argentine people they have a good national side. There's no doubt they'll come with everything so we have to be very concentrated," he told reporters.
"If we play against the big names and the [Argentina] shirt we're not going to do great things. We must liberate ourselves, the lads must forget those things so they can play a match they see as easy to win, going out [to play] with that mentality."
