The 38-year-old had apparently been earmarked as the Swans' leading replacement for Michael Duff, who was relieved of his duties on December 4 following a five-game winless run.
Tottenham are said to have turned down one approach from their Championship counterparts, but Swansea remained hopeful that they could prise Davies away from the Premier League.
Ange Postecoglou was questioned on the purported interest after Spurs beat Nottingham Forest 2-0 on Friday, but the Australian was in the dark about any potential update.
"No, not as far as I know. I'm sure if it progresses or there's any resolution we'll find out in the next few days," Postecoglou told reporters when asked if any breakthrough had been made in talks.
Now, a report from BBC Sport claims that Davies has turned down the chance to take the reins in Wales and will continue to work under Postecoglou's wing at Hotspur Way.
Davies was not the only name on Swansea's shortlist, but the second-tier side had made him their top candidate and must now scope out alternative options to the Tottenham assistant.
A former Wales youth international, Davies was previously on the books at Swansea City from 2010 to 2012, serving as assistant to Brendan Rodgers before following the Northern Irishman to Liverpool.
A brief stint in the Reading dugout under Brian McDermott followed for Davies, who subsequently reunited with Rodgers at Celtic and then Leicester City, helping the Foxes win the 2020-21 FA Cup.
The Watford-born coach departed the King Power Stadium alongside Rodgers in April, and he was out of work for a few months before being appointed as one of Postecoglou's lieutenants alongside Mile Jedinak, Ryan Mason and Matt Wells.
With Davies's help, Postecoglou took a record-breaking 26 points from his first 10 Premier League matches in charge of Tottenham, who have since dropped to fifth in the standings, four points adrift of leaders Liverpool.
Meanwhile, caretaker boss Alan Sheehan oversaw Swansea's 2-1 loss to Middlesbrough in Saturday's Championship affair, which leaves the Swans 18th in the second-tier table.