Vermaelen's back injury has forced Arsene Wenger into seeking defensive replacements - with the Belgian sidelined for an indefinite period - and Swansea's Williams is at the top of his wanted list.
The defender has long been on Arsenal's radar, but their interest previously cooled due to Swansea's asking price.
However, Wenger could now be forced to push through a move for Williams amid fears that Vermaelen faces a lengthy spell on the sidelines.
Vermaelen has been ruled out of the club's pre-season tour of Asia because of injury and it seems likely he will miss the start of the new Premier League.
Arsenal are short of options at the back with Sebastien Squillaci being released earlier this summer and Johan Djourou moving to Hamburg on loan.
It means Wenger could start the season with just two recognised central defenders in the shape of Laurent Koscielny and Per Mertesacker.
Williams is thought to be Arsenal's top target to bring in a new defender with Wenger keen to have four central defenders in his squad ahead of the new season.
Wales captain Williams has established himself as one of the best centre-backs in the country after two stand-out seasons in the Premier League with Swansea.
Swansea are bracing themselves for bid for Williams with the 28-year-old expected to jump at the chance of a switch to the Emirates Stadium and the chance to play in the Champions League.