La Albiceleste lost 1-0 to Germany in the World Cup final in Brazil four years ago and also suffered consecutive Copa America defeats to Chile in 2015 and 2016.
Messi, who turns 31 later this month, has won everything he can at club level but has only managed to win an Olympic gold medal with his country in 2008.
Now preparing for what could be his last World Cup, Messi carries the hopes of Argentina on his shoulders again and does so with intensified scrutiny on his performances, according to Zanetti.
"There is a lot of pressure on him, but not only in this competition but also in the past," Zanetti told Mediaset.
"Everyone expects it to be his World Cup and that Argentina can win, but it will be difficult because I see other more equipped national teams.
"The defeats in the two Copa America and World Cup finals further increase pressures on him and the team from the press. We expect a lot.
"[The World Cup] stops the whole country, there is great enthusiasm and passion for this national team and we hope to get to the end of the competition."
Argentina begin their World Cup campaign against debutants Iceland on Saturday, before taking on Croatia and Nigeria to complete their Group D fixtures.