Arsenal have had two offers turned down, including one of £40million plus £1 which they believed would trigger a sale clause.
With the summer transfer window closing at the end of August, the future of the Uruguayan remains uncertain with another month for the Gunners to consider increasing their bid or even for Real Madrid, reported to be interested, to make a move.
But Liverpool are digging their heels in, despite Suarez wanting to speak to Arsenal, with the 26-year-old fully involved in preparations for the new Premier League season that gets going on August 17.
"It's a story that can keep running but for me there is no new news," Rodgers told the Liverpool Echo.
"Luis is happy. He totally understands the club's point of view. He's in a difficult position in terms of not being able to play because of his suspension - he will sit out the first six games and he will miss playing football.
"But he's working hard and remains part of the group. He hasn't been moping about. It's obviously difficult when you come back after a break but he has settled in and has been working well.
"He needs to get up to speed and we will continue to work to get all the players ready for the first game of the season."
Suarez, who joined Liverpool from Ajax in January 2011, is banned for the first six games of the campaign but made his first appearance for Liverpool since biting Chelsea's Branislav Ivanovic's arm last season in last week's 2-0 win over Melbourne Victory in Australia.
He has also been on Confederations Cup duty for Uruguay and Rodgers has insisted his group of players have not been affected by the transfer speculation surrounding one of their star players.
"It could have unsettled them but it hasn't," Rodgers added. "It says a lot about their professionalism: the environment that we're creating is that the star will always be the team.
"We want to have top players here but if for whatever reason we don't, we will always know we can rely on the team and each other.
"The harmony in the group is very good. The spirit is strong. Right across the club there is a one club mentality. If we can keep it that way we can have a great season."