The Merseyside club were well off the mark last season after finishing in eighth spot, which was a primary factor behind the dismissal of Kenny Dalglish from Anfield.
"I've said right from the off I want to make us as competitive as we can possibly be," said Rodgers. "The challenge at the top of the table is greater than ever. Now there are seven or eight teams involved but the top four is certainly not something we'll shy away from.
"It's where a club of this status wants to be but it's always easier said than done and the reality over the past three seasons or so hasn't been that. Our job is to try to push on."
Rodgers will coach his first Premier League game in charge of Liverpool away to West Bromwich Albion tomorrow.