Arsenal made a late attempt to sign Fabinho before he joined Liverpool from Monaco, according to Sky sources.
The 24-year-old midfielder is admired by the Gunners' new boss Unai Emery, who tried to sign him at Paris Saint-Germain.
It is understood Arsenal made attempts to land the player over the weekend but he had already decided to join the losing Champions League finalists.
Fabinho will officially join Liverpool on July 1, after successfully completing a medical at Melwood on Monday.
Sky sources understand Liverpool will pay £39.3m (€45m) plus a maximum of £4.3m (€5m) in add-ons.
"I am really excited about this move," Fabinho told Liverpool's official website. "This is something that I always wanted - this is a giant of a team. The infrastructures of the club look exceptional.
"A football club of this size coming after my services, I didn't have to think that much about coming over. I will try to create my own history at this football club. Hopefully, on a personal level, I'm able to win titles with this club.
"I will try to grow and learn and improve myself, and to be part of the club's history."
Fabinho will join the club on the same day Naby Keita arrives from RB Leipzig.
Liverpool are also interested in Lyon forward Nabil Fekir, although the France international says he is "not close" to signing for the Reds and still needs time to think about his future.