Villas-Boas replaced Harry Redknapp as Spurs boss after chairman Daniel Levy sacked the 65-year-old despite the former Portsmouth manager guiding the club to fourth place last season.
The Hammers entertain Spurs on Monday night with Villas-Boas' side currently occupying fourth spot once more, but Allardyce said: "They've got to finish higher than fourth haven't they? That's the pressure for him."
He added: "He's done great hasn't he taking over from Harry? But fourth is the same as Harry and you know what we're like in football - they'll say 'Oh well, Harry done that'.
"I think they're good enough to finish third or fourth. I don't think there's any doubt about that. I think if they continue to maintain the top players fit they've got a really good chance and they might even get into that third spot. I was particularly impressed with them when we played them, their front line was outstanding."
Gareth Bale scored once with ex-Hammer Jermain Defoe bagging a brace as Spurs ran out 3-1 winners when the two sides met at White Hart Lane in November.
But Allardyce has overseen wins over the likes of Chelsea at Upton Park, where his side have also held Manchester United and Manchester City to draws.
And the 58-year-old believes home advantage could prove the difference on Monday.
"You can only be as good as your players are. If you've got great players you'll get the results and I think they've got some great players at Tottenham and they're really producing," he said.
"We know we've got a game on our hands on Monday but we're a different team at home and we enjoy playing against the big boys. It seems to bring the best out of our players and hopefully it will be the same on Monday."