Gunners boss Unai Emery is keen to strengthen his midfield options after losing Aaron Ramsey to Juventus.
Real Madrid paid around £16m to sign Ceballos from Real Betis two years.
But the 22-year-old has struggled to nail down a place in the Los Blancos first XI, playing just 35 La Liga games in two seasons.
And according to BBC journalist David Ornstein, Arsenal have beaten rivals Tottenham to Ceballos’ signature.
However, the deal is set to be a 12-month loan, which would see the Spain international return to the Bernabeu in 2020.
“Arsenal have reached an agreement to sign Real Madrid and Spain U21 midfielder Dani Ceballos on loan for 2019-20 campaign,” tweeted Ornstein.
“With William Saliba also set to complete move from Saint Etienne before returning there on loan next season, big week for #AFC in transfer market #RMFC #ASSE”
Arsenal are also set to sign Saint-Etienne centre-back William Saliba, 18, who is due for a medical at the Emirates imminently.
The France youth international will cost around £27m and will return to Saint-Etienne on loan next season before joining up with the Gunners in a year.
Saliba has expressed a desire to join Arsenal despite late interest from Tottenham.
Emery will now turn his attention back to Celtic left-back Kieran Tierney.
Arsenal are hopeful of raising funds by selling club captain Laurent Koscielny, who refused to travel on the club’s pre-season tour.
Celtic are reluctant to accept Arsenal’s £25m offer due to the way in which the payments would be structured.
The Scottish Premiership champions want a larger amount of the fee up front than the Gunners are currently offering.
Arsenal have already signed Brazilian attacker Gabriel Martinelli for around £6m.