Fikayo Tomori wants to stay at AC Milan after advanced talks with Tottenham, according to Sky Italy.
Spurs are looking to add a defender before Monday's 11pm transfer deadline and the 27-year-old was one of a number of options the club were looking at.
Postecoglou's side have been severely hindered by defensive injuries throughout the current campaign.
Postecoglou said on Friday Cristian Romero is still a few weeks away from a return, while fellow centre-back Radu Dragusin picked up an injury in Thursday's Europa League home win over Elfsborg.
Bayern Munich forward Mathys Tel has also rejected a move to Tottenham in the past 24 hours after the clubs had agreed a £50m deal.
Spurs are yet to sign an outfield player going into the final days of the transfer window.
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Speaking about the fitness of his back four ahead of the upcoming clash against Brentford, live on Sky Sports this Sunday, Postecoglou said: "Still not sure, we'll let it settle down and get more information over the weekend. He's [Dragusin] obviously not available for this weekend.
"In terms of others, Djed Spence should be available for the weekend, Sergio Reguilon should be OK.
"Every time I think I see the light at the end of the tunnel it turns out to be an oncoming train."
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