Wales' Ashley Williams
Toffees manager Koeman was a defender during his playing days, and he arrives at Goodison Park with Everton coming off a terrible season defensively, with only the bottom five teams conceding more goals than Everton in the Premier League last season.
Swansea's Ashley Williams looks dejected
Still, John Stones is attracting interest, and Koeman appears to have set his sights on a centre back to possibly replace Stones in Ashley Williams - but given Everton's problems at the back, he may well move for the Swansea star regardless of Stones' future.
The Mirror report that, whilst Swansea have rejected Everton's opening bid of £10m for star centre back Williams, Koeman is still desperate to land the Welshman before this month's transfer deadline.
Swansea's Ashley Williams celebrates scoring
Williams, who joined Swansea from Stockport back in 2008, and has racked up more than 350 appearances for the club whilst becoming one of the most dependable central defenders in the Premier League.
The report claims that Williams has two attributes that make him a top target for Koeman, as it's believed that the Everton boss sees Williams' leadership and passing ability as key for how he wants to re-shape his defence this summer.
Ashley Williams celebrates scoring for Swansea City
Koeman had a wily veteran in his defence at Southampton in Jose Fonte, and whilst he does have Phil Jagielka at Everton already, Williams appears to be the ideal choice for Koeman - and despite the fact that he turns 32 later this month, the Dutchman is believed to have no concerns about Williams' age.
Swansea may have stood firm after Everton's first bid, but with Koeman still pursuing Williams, we may well see the Toffees make an improved offer to try and finally land the man Koeman wants to spearhead his defence at Goodison Park.
Do you think Everton should make another bid for Williams?