Spurs are facing a Carabao Cup final against Manchester City on Sunday, their biggest chance to win their first trophy since 2008.
Woodgate was part of the Spurs side that won the club's last trophy, also the League Cup.
"They have to (start winning silverware) with the stadium they have built and with the players they have got," said current Bournemouth boss Woodgate.
"They have got the best striker in England in Harry Kane so they need to start winning trophies for him.
"They're a well-run football club but they need to start soon."