Villas-Boas was brought in to replace Harry Redknapp as boss last summer with the brief to get automatic qualification for next season’s Champions League.
And Allardyce, whose side take on Spurs at Upton Park tonight, said: “They’ve got to finish higher than fourth, haven’t they? That’s the pressure for him.
“Villas-Boas has done great taking over from Harry. But fourth is the same as Harry. And you know what happens in football — they’ll say ‘Oh, well Harry done that’.
“I think Tottenham are good enough to finish third or fourth as long as they continue to keep the top players fit and don’t pick up too many injuries.
“If they can do that, they’ve got a really good chance of getting that third place.”