Danso, 27, made his first appearance of the season in Wednesday's Carabao Cup win over Charlton, introduced from the bench in the second half.
The Austria international's route to regular league football looks increasingly difficult after Tottenham strengthened their central defence with the signings of Marcos Senesi and Jan Paul van Hecke, while Micky van de Ven has signed a new long-term contract and been confirmed as club captain.
Sunderland's approach, revealed by BBC Sport, was for a season-long loan, and it remains to be seen whether the Black Cats renew their interest before the transfer window closes.
Danso joined Tottenham on loan from Lens in January 2025 before the move was made permanent later that year for a fee of £20.9m; he has made 37 appearances for the club since arriving.
Elsewhere, De Zerbi offered an encouraging update on Dejan Kulusevski, who has been sidelined for 15 months with a complicated knee injury. The 26-year-old is being phased back into full training with the group, and football.london reports he is being handled carefully by staff, with no pressure being applied over a timeline, though there is growing positivity around his recovery.
De Zerbi, who must finalise his Premier League squad list before the window shuts next week, flagged concerns over his attacking options, with Wilson Odobert and Mohammed Kudus both working their way back from injury.
"Odobert is injured for a long time. Before February, March, he is not available, and then maybe one of these [attacking] players can leave before the end of the window. I think we have not too many players," De Zerbi said.
"And also if you consider last season, not for Tottenham, the Premier League … all clubs suffered many injuries, no? And we have Kudus after six months injured.
"I will be very, very happy to see Kulusevski in [the squad list]. I want to put Kulusevski on the list because he is working very well, very hard. He's a good guy and he's a different player. Xavi Simons … before March, April, he's not ready. So I think we need players; this number of players."
On Kulusevski specifically, De Zerbi added: "If he feels good like now [he'll be in the squad list], he's improving day by day. It's great watching him during the season and also because he deserves it because he suffered too much for his career, for his age, for his level of human. I'm not speaking of him as a player but as a human. He deserves to stay in the list."