The Spain midfielder has impressed in his first four Gunners appearances since signing on loan from the LaLiga giants in July.
In his full debut for the Gunners, Ceballos registered two assists and picked up the Man of the Match award in a 2-1 win over Burnley last month.
His early-season performances saw him start in Spain’s 2-1 victory against Romania on Thursday.
Football.London claim Arsenal wanted an option-to-buy in the loan deal, but Real Madrid refused their proposal.
However, the report states Real’s stance could change if manager Zinedine Zidane leaves the club.
The Frenchman has endured a difficult start to the 2019/20 campaign, winning just one of his opening three games.
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And a run of bad results could result in Zidane’s departure after spending more than £250million on summer signings Eden Hazard, Luka Jovic, Ferland Mendy, Eder Militao, and Rodrygo.
Ceballos further endeared himself to Arsenal fans by passionately celebrating Alexandre Lacazette’s goal from the bench in the club’s 2-2 draw with Tottenham last weekend.
Arsenal right-back Hector Bellerin praised Ceballos’ arrival at the Emirates.
He told the club’s website: “I think Dani’s one of the best players I’ve seen in my career.