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.
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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