A move to Liverpool's rivals still hinges on Gakpo's own club first landing a replacement, with talks over a £120m deal for Paris Saint-Germain's Bradley Barcola having advanced significantly this week.
Andoni Iraola wants more than one new winger before the window closes regardless, and the London Evening Standard reports Liverpool have made checks on Crystal Palace's Ismaila Sarr as a further option.
Even if Liverpool land both of their targets, they are still expected to hold out for a substantial fee for Gakpo, who has four years left on his contract.
De Zerbi confirmed the club's intentions to reporters on Friday.
"I didn't expect a transfer window like we have done," he said. "I think we can do one player more in one specific position. Yesterday we went to dinner with the owners, Johan [Lange, sporting director] and Vinai [Venkatesham, chief executive] and we spoke about this. All of us are on the same page and a very good connection."
Tottenham are expected to remain busy through to Tuesday's 11pm deadline, both in terms of incomings and departures.
"I've been very clear with all the players: if you speak clearly you can't expect surprises," De Zerbi said.
"The players who wanted to leave have left. We maybe have some players who want to leave and they are waiting, we are waiting for an offer. You have to feel proud to stay here. It's an emotional project we are building with new players."