Argentina host fourth-placed Peru on Friday (NZT), before travelling to Ecuador five days later and need at least a win and a draw to be safe.
"The pressure on the players is going to be incredible," said Goycochea. "The pressure will be so heavy, and that's the concern. It's hard to play under that kind of expectation. Before these qualifiers, everybody assumed Argentina would be there. They have everything to lose and nothing to gain."
Argentina is laden with stars. Apart from Messi, they also boast Sergio Aguero, Angel Di Maria and Paulo Dybala, who has scored 10 goals in six games for Juventus and is valued at more than 90m ($163m). Surely they are favourites to beat Peru?
"Yes, we have a better team than Peru," said Goycochea. "But we also have a better team than Venezuela - and we couldn't beat them. That's the way football is."
Goycochea, best known for his penalty saving heroics which prompted Argentina's unlikely passage through the 1990 World Cup, still expects Argentina to qualify but admits it will be a nervous wait.
"There is a lot of anxiety here," said Goycochea. "Everybody is nervous ... because it is football and anything can happen."
That view is echoed by Sebastian Garcia, a football writer with mundoalbiceleste.com
"We are pretty much panicking," said Garcia. "My heart says we will make it, but my head is worried. If someone offered me now the guarantee of a playoff spot against New Zealand, I would sign it in blood."
Garcia says it has been a crazy campaign, with three different coaches in charge and a series of unforeseen results.
"In the past, it has been practically automatic that we win all our home games and compete well away," said Garcia. "This time, we have lost to Ecuador and Paraguay, and couldn't beat Venezuela at home. At the last World Cup, we were in the final, and now we might not even qualify. Our team hasn't gelled, they don't seem to click together."
Argentina have scored just 16 goals in 16 games but Garcia would feel confident about facing New Zealand in November.
"As long as you don't field the All Blacks as well, we should be okay," said Garcia. "I would take that option right now ... because the danger is Argentina could miss out altogether.
"That would be a disaster; the only thing we have left here is football. The economy is a mess, unemployment is racing and there is lot of instability. The only thing we are united about is football and the national team so if they fail ... I can't imagine."