Nottingham Forest have had a £38m bid for the Sweden midfielder rejected and are preparing an improved £45m offer. De Zerbi addressed the situation before a pre-season friendly against Auckland FC.
"I was very clear at the start of my time in Tottenham. I said, who doesn't want to stay in Tottenham? Who is not happy, is not proud to stay here, has to leave," De Zerbi said. "I want the players to be proud and happy to stay here, to go on the pitch in Tottenham Hotspur Stadium with big motivation.
"Because now in football, if you think also in the World Cup, the motivation makes the difference. The motivation is to be happy to stay in one team."
Bergvall's game time has shrunk since De Zerbi's arrival in March. Spurs have since paid £100m for Sandro Tonali and £85m for Mateus Fernandes, part of a summer outlay past £200m alongside Jan Paul van Hecke, Martin Dubravka, Marcos Senesi and Andrew Robertson.
"I have not spoken with Lucas yet," De Zerbi said. "I will in the next days and we will make the best decision for the club, for Lucas, for me, for every one of us, without problems. Lucas is a big, big talent. There will be a big competition to play because at the end, I think we will be two players for each position. If he wants to stay, I will be happy. Otherwise, we have to find a new solution with Johan Lange, with Vinai. It's not my business after the decision to leave."
De Zerbi confirmed the focus has shifted to attack, with Spurs linked to Eli Junior Kroupi, Savinho, Pedro Neto and Rafael Leao.
"First of all, I think it's not finished yet, our transfer market window," he said. "We have to finish, we have to bring another couple of, maybe more, maybe not less, but a couple of players. The target now is to complete the attacking positions. We want the first target, the first level, because anyway we are already important players, but I think to start the new project we need another couple of important players, for sure."
Mohammed Kudus and Dejan Kulusevski remain absent through injury.
"Kudus is very close. I think when we come back to London he will be available to start working with us," De Zerbi said. "Kulusevski, we have to see in the next couple of weeks. After one season, one complete season, without a game, we have to analyse better and better, don't take risks, first of all for him and then for us."
Elsewhere, Xavi Simons has spoken about the ACL injury that ruled him out of the World Cup.
"The first four days were the worst days of my life," the Netherlands international told the club's website. "I couldn't sleep, I didn't eat for one week. I was in a lot of pain. I was vomiting and felt dizzy, it was everything. That pain, I don't wish no-one to go through that pain. This is what it is. I'm happy and in a good place now."