Suarez has been the subject of two bids from Arsenal, the latest a £40million-plus-£1 offer triggering a clause in the player's contract where the club were duty bound to inform him of the offer.
Suarez, 26, came on as a 78th minute substitute at the Melbourne Cricket Ground in front of a sell-out 95,000 crowd in a game which saw goals from Reds captain Steven Gerrard and Iago Aspas secure a 2-0 win.
The Uruguayan created the second for Liverpool's summer recruit from Celta Vigo and the Liverpool manager demanded some integrity from the South American star who has also been linked to Real Madrid this summer.
He had been allowed to join up with Rodgers' squad in Australia after Confederations Cup exertions in Brazil and Rodgers said: "There's nothing new to report. He's very much a Liverpool player and over the course of the next couple of weeks we've got to get him up to speed.
"The support he has received from the supporters and the people of the city of Liverpool has been unrivalled. In this period of time he's missed a lot of games for the club through various reasons. The people have stood by him, like a son and really looked after him.
"Whatever happens in the coming weeks that will be in his mind because it's something you can never forget."
Suarez still has six Premier League games to sit out, of a 10-match suspension, after being found guilty of biting Chelsea's Branislav Ivanovic.