Chelsea have also been monitoring the form of the £30m-rated defender, who has been a key figure for one of the surprise packets of the Premier League season.
Both the champions and the Gunners are looking at strengthening their defences in the transfer window.
Arsenal's need was further illustrated after they conceded three goals in the draw with Liverpool on Friday.
Blues boss Antonio Conte is a fan of Southampton's Virgil van Dijk while Gunners chief Arsene Wenger has targeted £30m David Luiz, who is out of favour at Stamford Bridge.
But Tarkowski is understood to be seen as a possible alternative next month.
Following the sale of Michael Keane for £30m in the summer, Burnley boss Sean Dyche turned to Tarkowski and the 25-year-old has been a big part of their success story this season. In his last nine Premier League appearances for them, the Clarets have conceded just three goals.
Burnley paid just £3.6m for Manchester-born Tarkowski when they signed him from Brentford in February of last year and he has 18 months left on his deal.
There is no suggesting he is angling for a move and Dyche would be reluctant to let him go.
Tarkowski served the first game of a three-match ban when Burnley lost 3-0 to Spurs.
He also misses trips to Manchester United today and Huddersfield on Saturday, having been punished for elbowing Brighton's Glenn Murray.
England boss Gareth Southgate is keeping an eye on Tarkowski, whose paternal grandfather left Poland for England after the Second World War.
Poland have made inquiries about him, but he has yet to give any indication who he favours.