Manuel Pellegrini is happy to see Adrian making an immediate impact at Liverpool but would have liked to have kept the goalkeeper at West Ham.
Adrian left London Stadium as a free agent at the end of last season and signed for European champion Liverpool, replacing Simon Mignolet as Alisson's back-up.
But an injury to Alisson in the Premier League opener against Norwich City saw Adrian quickly thrust into the action at Anfield.
And on just his first start for Liverpool, the Spaniard became the UEFA Super Cup final hero with a decisive penalty shoot-out save from Chelsea's Tammy Abraham on Wednesday.
Adrian had been restricted to cup appearances last season, with Lukasz Fabianski the West Ham number one, but Pellegrini still sought to keep the 32-year-old.
"I'm happy for him because he played a lot of years here at West Ham," Pellegrini told a news conference, reflecting on Adrian's display against Chelsea. "He's a very good goalkeeper.
"For different reasons, he couldn't find an arrangement to continue here. We wanted him here because not only is he a good goalkeeper, he's a good player for the squad also.
"He had some problems at the start of the season, so signing for Liverpool and having this performance, I'm very happy for him."
Adrian spent six years at West Ham, largely operating as the club's first choice prior to Fabianski's 2018 arrival.
The former Real Betis man is set to continue in the Liverpool goal against Southampton this weekend, with Alisson out for "the next few weeks" - as confirmed by Jurgen Klopp.