Rodgers took over as the Reds boss last summer, and Alonso, who played at Anfield from 2004 to 2009, believes that they are currently a work-in-progress under their new manager.
"With Brendan, Liverpool are building a project, they need to be patient," Alonso told the BBC.
The Real Madrid midfielder also admitted that he considers himself to be a Liverpool fan after winning the 2005 Champions League and the 2006 FA Cup with the Reds.
"I still follow the Premier League and I feel very attached to Liverpool. I follow them and I consider myself a fan.
"We all want Liverpool to be challenging for the Champions League places."
The Spain international joined Madrid from the Reds for £30m in 2009.