Donczew has been absent from Cardiff's pre-season preparations this summer after attracting interest from several Premier League clubs, with Fulham previously credited with an interest before Arsenal moved ahead in the race.
The Telegraph had first reported that Arsenal were showing strong interest in the youngster, with the expectation that he would initially join the club's academy setup rather than push straight for first-team involvement.
Donczew made history at Cardiff last October when he came on as a second-half substitute against Newport County in the EFL Trophy at 15 years and 234 days old, becoming the club's youngest-ever player.
He added a second senior appearance a fortnight later in an EFL Trophy tie against AFC Wimbledon, and was later named among the substitutes for Cardiff's EFL Cup quarter-final against Chelsea, though he remained unused on that occasion.
Cardiff manager Brian Barry-Murphy, who previously spent three years leading Manchester City's Elite Development Squad, has been effusive in his praise of the teenager.
"I think if they're good enough, and he clearly is, then they're old enough for us with Axel," he told WalesOnline of Donczew back in December.
A right-footed left winger capable of playing more centrally as an attacking midfielder, Donczew is already a Wales under-19 international despite his age, having represented the country across several youth age groups with 11 appearances in total. He remains eligible to play for Poland, and Wales senior head coach Craig Bellamy is said to be an admirer.
Cardiff's under-21s coach Darren Purse said Bellamy's early enthusiasm for the player, as an Under-16 with Wales, played a part in shaping Barry-Murphy's own view of him.
News of the Donczew deal comes less than 24 hours after it emerged that Arsenal are exploring a move for Real Madrid forward Vinícius Júnior, with reports suggesting the north London club has internal approval to continue pursuing the Brazil international this summer, underlining the scale of Arsenal's ambitions at both ends of the transfer market this window.