The attacking midfielder has been blighted by hamstring troubles so far this season, but could be set for a return in Thursday's Europa League clash with the Serie A side
The former England international has not featured for the Reds since suffering a recurrence of a hamstring injury in the 3-0 defeat to West Brom on the opening day of the Premier League season.
"His attitude is superb and Joe really thinks he can contribute to the group, which is great," Rodgers told the club's official website.
"He's been unfortunate. He's been out and came back a couple of times, but didn't quite really get through fitness wise.
"He was out for eight weeks, trained for a couple of weeks, had his first game, scored a good goal and now he is fit. He's back into training now and looking to fight for a place in the squad.
"He loves his football, Joe, and he's now fit enough to fight for his position in the group."
Lucas Leiva and Martin Kelly remain the only long-term injury worries, while Spanish full-back Jose Enrique is close to a return to action.
"We're fine. We're pretty much the same as we were at the weekend. The boys that are a bit longer-term like Lucas Leiva are still working well,” said Rodgers at a pre-match press conference.
"There's nothing new from the weekend. Jose Enrique is back out on the pitches running. Apart from that, everyone is good and the condition of the team is good. We'll look forward to tomorrow night."
Though Rodgers fielded a young and inexperienced side in Liverpool’s 5-3 win over Young Boys, he is expected to start a number of more senior players, including Steven Gerrard and Luis Suarez, against Udinese.
"I always look at each game and pick a group and team to win it," he continued.
"We need to ensure we have a group that is strong enough to win the game, but those players [Gerrard and Suarez] will be involved."