It is believed Spurs have bid around £8 million for the left-back, who has enjoyed a breathtaking rise to prominence since breaking into the Swansea first team less than two years ago.
Having been drafted in as cover when Neil Taylor suffered a serious ankle injury in September 2012, Davies excelled, keeping his place when the former Wrexham man returned to fitness and throughout last season.
Davies has established himself as a Wales international during that period and was voted young player of the year for the previous two campaigns.
It is thought Spurs are demanding £10 million for Sigurdsson, who spent a successful spell on loan at Swansea under Brendan Rodgers in 2012.
And it could be that Swansea accept the Icelander in a straight swap for Davies.
But even that might not satisfy the Liberty hierarchy because Luke Shaw's bumper £30 million switch from Southampton to Manchester United has somewhat skewed the market.
Shaw is just two years and three months younger than Davies and has made fewer Premier League appearances, so, as highly rated as the England international is, Swansea may question why their player is worth just a third of that price.
Davies has another three years to run on the contract he signed last term, meaning Swansea are under no pressure to sell.
If their valuation is met, it may make sense to let Davies go as they have a ready-made replacement in Taylor.
Though Davies has edged the argument in the last two campaigns, there is little to choose between the pair.
Taylor is highly thought of at Swansea and featured regularly in Europa League games last season, captaining the side on more than one occasion.
Monk is unlikely to have a problem turning to a player who represented Team GB at the Olympics just two years ago.
And the suggestion is the 25-year-old may look to move on himself if Swansea cannot guarantee him more game-time than last season.
Taylor has spoken of his frustration at spending so much time on the bench, while his agent, Jamie Moralee, insisted another campaign as a bit-part player would be unacceptable to his client.
Should Taylor leave, Swansea are unlikely to get anywhere near the fee on offer for Davies.
And if a deal can be struck that would bring Sigurdsson back to the Liberty, it may represent a shrewd piece of business.
The attacking midfielder made a big impression during his first stint in SA1, scoring seven goals in 18 Premier League appearances.
Swansea were close to bringing him back to South Wales on a permanent basis two years ago, but Brendan Rodgers's departure to Liverpool, coupled with a bumper offer from Tottenham, saw the former Hoffenheim man head for White Hart Lane. Sigurdsson endured a slow start to life in North London, though he played regularly and ended 2012-2013 with a flurry of goals.
He scored a respectable five times in 27 top-flight appearances last term, but a change of manager at Spurs, following Mauricio Pochettino's summer appointment, might affect his situation.
Swansea's interest in Sigurdsson casts further doubt on Michu's Liberty future.
The ex-Rayo Vallecano player was Michael Laudrup's first signing, bought as a replacement after Swansea lost out on Sigurdsson to Spurs.
Both players are best suited to operating in an advanced midfield role behind a main striker, and Michu has regularly been linked with a move away following Laudrup's departure.
Swansea have rubbished weekend reports that their 2012-13 top scorer has undergone a medical at Napoli.
West Ham are among others said to be tracking a player who was capped by Spain last October and is still likely to be sought-after despite an injury-hit campaign.