The Catalan giant is said to have been focused on handing out bumper deals to its most promising talents after losing Lionel Messi in pre-season, with a rebuild well under way.
Media speculation in recent days suggested their hopes of renewing Fati's deal had hit a stumbling block, with his agent Jorge Mendes apparently unwilling for a €1billion release clause to be inserted into his contract.
Such a clause proved no obstacle to securing Pedri's future and quickly the news reports were turned on their heads, with Mundo Deportivo claiming on Wednesday that a deal has been struck with Fati and Mendes "in principle".
Fati has broken a host of records since his emergence as a 16-year-old in 2019, including becoming Barcelona's youngest ever LaLiga goalscorer, and has looked sharp since returning from a serious knee injury.
He has already scored twice in just 94 minutes of LaLiga action this term, with Koeman easing him back.
Wearing Messi's former number 10 jersey, he looks set to lead Barca for at least the next five years.
"It's true that they are close [to an agreement]," Koeman said.
"It's very important news. He's a young player with a lot of quality. He's one of those who should shape the future of this club."
Fati's previous deal was technically due to expire next year, though Barca did have the option to unilaterally extend the agreement until 2024, much like the case with Pedri prior to the signing of his new contract.
Pedri signed on until 2026 and Fati is rumoured to be in line for a similarly lengthy extension.