Arsenal have had two bids rejected, the last one of £40million plus £1 which they believed would trigger a sale clause.
Liverpool have always maintained they merely had to inform the player of offers of that magnitude and have been steadfast in stating Suárez is not for sale.
They believe they are owed a degree of loyalty from the 26-year-old after standing by him during the race row with Patrice Evra which saw him banned for eight matches and during his current suspension of 10 games for biting Chelsea's Branislav Ivanović.
The 26-year-old Uruguay international still appears keen to leave Anfield, although he has stopped short of submitting a transfer request or seeking legal recourse, but Rodgers said it had not affected his attitude.
"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."
Suárez's situation was the transfer saga of the summer until it was recently overtaken by Real Madrid's pursuit of Tottenham's Gareth Bale with hardly a day going by without him being in the headlines.