The Wales international has been a long-term target for the Gunners, but Reds boss Brendan Rodgers is thought to be keen to bring his former-player to Anfield.
The Liverpool manager returned to Swansea to sign Joe Allen for £15m last summer, and is thought to rate Williams very highly.
The Swans defender has been valued at £8m, according to a similar report from the London Evening Standard, and the 28-year-old's stock has risen in the last 18 months thanks to his impressive performances since stepping up to the Premier League.
The report claims both clubs are ready to move for the defender in the summer. Rodgers agreed not to bid for any of Swansea's players for a year following the signing of Allen, but the Reds manager is tempted to go back for Williams once that agreement expires at the end of the season.
But Arsene Wenger is also a big fan of the Welsh centre-back, and is reportedly eager to strengthen his defence. The Gunners beat West Ham 5-1 at the Emirates Stadium last night, but their old defensive frailties caused concern once again in the first-half, as the Hammers went 1-0 ahead before Arsenal hit back.
Williams agreed a new three-year contract with the Swans last October, so Swansea will be in a strong position when it comes to negotiations should any bids be forthcoming.