The Denmark international has been the subject of reported interest from Barcelona, who are said to have had a £15m bid rejected by the Merseyside outfit this week.
"The job of the captain and vice-captain here at Liverpool is very, very important," Rodgers told the club's official website. "I wanted to give myself time over the course of the pre-season to assess a number of the players and obviously Daniel was one of them.
"His loyalty and his passion for Liverpool is second to none. And if you add that on to his footballing qualities, then I think it's a good choice for everybody."
The 28-year-old has made 209 appearances, netting 12 goals, since moving to Anfield from Brondby in 2006.