The Belgium international can leave Tottenham this summer in a cut-price £25m deal.
But any move must be completed before the final two weeks of the summer transfer window.
Alderweireld has been heavily linked with a move to Manchester United over the last few months.
The Red Devils are in desperate need of defensive reinforcements for the start of next season.
But many of Europe’s top sides are also eyeing the 30-year-old at the cut price.
And, according to The Sun, Arsenal and Chelsea have now entered the race for his signature.
Manchester City too are monitoring Alderweireld’s situation and could be tempted to bid for the centre-back.
Unai Emery is aiming to complete several bargain deals this summer as he remains in the dark about his transfer budget.
Elsewhere at Arsenal:
“Arsenal eyeing shock Tottenham raid for Man Utd transfer target – fans will NOT be happy”
Arsenal may have just £40m to spend if Emery fails to deliver Champions League football this season.
While Tottenham are prepared for Alderweireld’s exit this summer, they are yet to give up on convincing the him to sign a new contract.
Chelsea’s interest in Alderweireld may be short-lived as the Blues wait to learn their fate in regards to their transfer ban.
They could be blocked from signing any players for the next two windows if their appeal against the ban is unsuccessful.
Arsenal meanwhile are preparing for their Premier League clash with Watford on Monday night.
Emery must decide what team to put out onto the pitch with a Europa League quarter-final second leg at Napoli to come on Thursday.
But the Gunners still have work to do in the Premier League this season with a place in the top four still up for grabs.
Going into the game with the Hornets the team are just outside the Champions League spots.