The Gunners are in the market for a long-term successor to Petr Cech, who turned 36 last month.
And according to Sport, Unai Emery has asked Arsenals hierarchy to sign a new goalkeeper this summer.
The report claims that Arsenal have made a direct approach to Barcelona to gauge whether a deal is possible.
But Chelsea have also enquired about Cillessens availability due to the uncertainty over Thibaut Courtois future.
Barcelona are ready to stand firm, however, and are not prepared to sell the Netherlands international for anything lower than his €60 million (£52m) release clause.
Cillessen is currently second choice to Marc-Andre ter Stegen but is understood to be willing to remain at Barcelona unless he receives an offer he cant reject.
Liverpool have also been linked with a move for the 29-year-old but its Arsenal and Chelsea who have made the first moves.
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