The Reds finished seventh in the Premier League last season, 12 points behind Arsenal, who claimed the final Champions League place.
However, Alonso feels that Rodgers's methods started to bear fruit towards the end of the season and thinks that the Northern Irishman will guide the club back to the competition they have won five times.
"We trust that in the next years, the team will come together, become more competitive and reach the Champions League," Alonso told estoesanfield.com.
"Maybe he found it a bit harder than he expected, but at the end of the season with the new signings the team played quite well.
"When Brendan arrived we knew it was a project for a few years, and step by step he started the season trying to establish his methods."
The Spaniard also revealed his love for the club he played for from 2004 to 2009 and said that he feels part of the club's illustrious history, having won the Champions League and the FA Cup during his time at Anfield.
"I enjoyed five wonderful years, we had great nights and great games. We achieved a Champions League [win] and I think that will remain in the history forever, so I was a little part of [their] history," he added.
Alonso was sold to Madrid for £30m.