The Premiership outfit are already showing serious interest in Hoops central defender Virgil van Dijk. Now they are preparing to launch a £2million bid for full-back Matthews.
He saw his childhood dream of signing for hometown club the Swans shattered when he broke an arm while training with them as a schoolboy. Matthews eventually signed for local rivals Cardiff, where he became a first-team player in his teens.
Swansea boss Gary Monk is targeting three new signings to revitalise a side which struggled last season and Matthews and van Dijk have been identified as two of them.
With Sunderland and Southampton also in the hunt for the Dutchman, Van Dijk's asking price is likely to be pushed well beyond what Swansea want to pay.
But they are hopeful they can entice Celtic to part with 22-year-old Matthews for a knock-down fee.
That's not expected to even be given consideration by the Parkhead club, who brought Matthews north on a Bosman in 2011, and who have him safely under contract for another two years.
New Hoops boss Ronny Deila is understood to be ready to allow some of the squad he inherited from Neil Lennon to move as they are surplus to his requirements.
And, if Celtic crash out of the race for a Champions League group Stage place when they play Legia Warsaw in the second leg of their third qualifying round tie on Wednesday, the new manager may be forced to lose some of the biggest stars in his squad, including Fraser Forster and van Dijk.
However, Matthews is a player not on Deila's "available" list.
So any overtures from Swansea will be unwelcome - certainly if the offer is only £2m.
Deila made a surprise call when he played Matthews at left back ahead of Emilio Izaguirre for the 4-1 defeat by Legia in Warsaw last week.
Despite the bitterly disappointing result, it was more confirmation that the Norwegian believes the man who has won 12 caps for Wales has the qualities he wants in his team.
Deila is looking to strengthen his squad, rather than let any of the players he rates leave.
His ability to spend will be enhanced if he can turn around their qualifier against Legia and make it into Friday's draw for the playoff round - and if he can generate income from the sale of players he has adjudged will not be part of his plans.
Tony Watt has already been offloaded for £1.2m to Standard Liege, where he made his debut at the weekend, winning a penalty for the winner when he came off the bench.
And Amido Balde could be set to join the striker in Belgian football - where he previously played for Cercle Brugge -with Lokeren hoping to end his unproductive time at Celtic.
However, Celtic are eager to recoup as much as possible of the £1.8m they paid Vitoria Guimaraes for the 23-year-old last summer.
With the two clubs a distance apart in their valuations, Balde - who came on as second-half sub in the 6-1 hammering by Spurs in Helsinki at the weekend - looks to be going nowhere for now.