The 20-year-old Sweden international has been linked with a move away from Spurs, with Newcastle United and Nottingham Forest both reportedly interested in signing the midfielder.
Bergvall's future has come under increased scrutiny following Tottenham's big-money arrivals of Matheus Fernandes from West Ham United and Sandro Tonali from Newcastle, increasing competition in midfield.
Despite the speculation, De Zerbi said Bergvall has not told him he wants to leave. The young midfielder started Tottenham's pre-season friendly against Sydney FC on Wednesday, with Spurs eventually winning 4-2 on penalties after the match ended 1-1.
"I've always been very clear with him and with you. I would like him to stay with us," De Zerbi said.
"I think he has the potential to become a top, top player. But I don't want to convince anyone to stay.
"Tottenham is already a top club without me, and we don't have to convince anyone to stay.
"I can't guarantee a place in the first XI to any player in the world."
Mathys Tel gave Tottenham the lead with a superb free-kick before half-time, but Sydney hit back through new Japanese signing Takahiro Sekine to force a draw before Spurs won the penalty shootout.
The result followed Tottenham's 2-0 victory over Auckland FC as De Zerbi's side continue their preparations for the new Premier League season, which begins against Brentford on August 22.
Reflecting on the performance, De Zerbi was pleased with his team's display, particularly before the interval.
"I think we played well, especially in the first half," he said.
"In the second half we changed many players, but the quality of the football remained good. We are working very well with the younger players."
New signing Sandro Tonali made his first appearance of the tour after missing the opening match with a minor injury, playing the entire second half. Fellow recruit Andy Robertson also made his debut, coming on for the final 25 minutes.
However, Matheus Fernandes did not feature, with De Zerbi revealing the midfielder was rested as a precaution because of a calf issue.
"It was just fatigue in the calf and we decided not to take any risks," he said.
The Italian also suggested Fernandes is unlikely to face Chelsea in Tottenham's final tour match in Sydney on Saturday.
Several senior players, including James Maddison, Kevin Danso, Destiny Udogie, Micky van de Ven, Jan Paul van Hecke and Dominic Solanke, were also unavailable.
"Danso, Van de Ven, Solanke and Jan Paul van Hecke were not ready to play without risk," De Zerbi said.
"Maybe some of them can play on Saturday against Chelsea. Maybe not from the beginning, but perhaps Robertson and Sandro Tonali will start.
"It is about managing the players. I wouldn't like to take any risks."
Tottenham's next match is a pre-season Sydney Super Cup friendly against Chelsea on Saturday at Accor Stadium in Sydney, Australia.
Jack Grealish has dismissed suggestions that he was dropped from Manchester City's pre-season tour, insisting he is still recovering from injury rather than being left out by head coach Enzo Maresca.
Chelsea have completed the signing of defender Maxence Lacroix from Crystal Palace in a deal reported to be worth 52 million pounds.
Newcastle United have decided to amicably part ways with manager Eddie Howe just a few weeks ahead of the upcoming 2026/27 campaign.
Liverpool maintained their winning start under new head coach Andoni Iraola after teenage forward Rio Ngumoha scored the only goal in a 1-0 pre-season victory over Wrexham at New York's Yankee Stadium on Wednesday.
