The Internacional player, who has won 14 caps for his country, was Spurs' main target of the January transfer window.
Villas-Boas was keen to reinforce a forward line weakened by Jermain Defoe's ongoing abdominal injury and the loss of Emmanuel Adebayor to the Africa Cup of Nations.
An offer in the region of £13m was rejected on deadline day and Villas-Boas said: "We've approached this market very sensibly.
"I've said in our conversations that if (the Damiao bid) was going to go out of proportion, we would step out of it.
"We have made it public before that he is a player that we chase, but we couldn't agree on the full conditions so we've moved on."
Villas-Boas did concede being left with two strikers is a risk, but brushed off suggestions the club should have pushed the boat out to sign Damiao.
Tottenham, who have a club partnership with Internacional, were expected to renew their interest in Damiao this summer, but Villas-Boas claims that is now unlikely.
"No I don't think (it will happen in the summer)," he said. "It's probably finished now."
To find out more about live football on Sky Sports, Click here