Spurs finished 17th in each of the past two Premier League seasons and narrowly avoided relegation on the final day of last term. De Zerbi arrived in March after Thomas Frank's difficult spell and an interim period under Igor Tudor.
Tottenham were bottom and winless at the first international break, but three successive victories have lifted them to ninth before Saturday's trip to Leeds.
Speaking on The Sports Agents podcast, Venkatesham said the club's ultimate objective was to win major honours.
“If you work for Tottenham, when we talk about being here to win, that means we're here to win the Premier League. That's what this football club wants to do.
“It wants to win the Premier League and it wants to be consistent in the Champions League and it wants to ultimately win the Champions League.
“I want them to never settle and I want us to be completely relentless in that pursuit.”
### Venkatesham reflects on difficult season
The ambition is a contrast with Tottenham's position during the closing weeks of last season. De Zerbi's side beat Everton 1-0 on the final day to stay in the Premier League, sending London rivals West Ham down.
Venkatesham admitted the period had been extremely challenging and confirmed that the club had prepared for the possibility of relegation.
“It was really challenging, a really, really challenging period for the club,” he said.
Asked whether he had made a plan in case Tottenham went down, he replied: “Yeah, you have to. You can't not.
“To not make that plan would have been irresponsible. When I look back at that period, I take probably two things.
“I think our supporters were unbelievable. The last six or seven games, when they had faced probably the worst season they have ever faced, they got completely behind the team and the head coach and dragged us over the line. This is not normal.
“The second thing is sometimes going through something as traumatic as we went through last season bonds people. It puts even more fire in everyone's belly to make sure something like that never happens again.”
Venkatesham's message is that survival cannot be Tottenham's ambition under De Zerbi. The club wants to return to the top of English football and establish itself as a consistent presence in the Champions League.