The Italian champions are understood to be keen on bringing the Polish shot-stopper to Turin as backup to Gianluigi Buffon initially, but with a view to him taking over the number one jersey if Buffon retires next summer as expected.
Szczesny has spent the past two seasons on loan at Juve's Serie A title rivals Roma, keeping 14 clean sheets in 38 league appearances last term.
The Daily Mail reports that the 27-year-old is eager to complete the switch to the Champions League runners-up as soon as possible, with a £13m deal thought to have been agreed between the two clubs.
Szczesny was left out of Arsenal's squad for their pre-season tour of Australia in order to finalise the move, which will come one year before the end of his contract at the Emirates Stadium.
The Gunners could also lose backup goalkeeper David Ospina this summer, with Fenerbahce interested in a £4m transfer.