The 22-year-old has made 56 appearances for the Spanish giants since joining in 2017 from Real Betis.
Gunners coach Unai Emery said:
“We’re excited to see Dani join us. He is a talented player with big technical ability, creativity and precision.”
Arsenal are also set to sign Saint-Etienne centre-back William Saliba, 18, who is expected to cost about £27m.
The France youth international will return to the Ligue 1 club on loan next season before joining up with the Gunners in 2020.
Saliba has expressed a desire to join Arsenal despite late interest from Tottenham.