The Uruguayan international had voiced his wish to leave Anfield at the start of the summer and he has since been linked with a move to Real Madrid.
Liverpool’s Premier League rivals Arsenal have had a £30 million (Dh169.1m) offer rejected and then had a second bid of £40m and a pound turned down.
The London’s side second bid has triggered a clause in Suarez’s contract that forces Liverpool to let him know of the bid and allow him to hold talks with the London side if he wishes to do so, though the Merseyside club are not under any obligation to sell the player until they are offered a fee that meets their valuation.
Suarez on Wednesday made his first appearance for Liverpool since biting the arm of the Chelsea defender Branislav Ivanovic last season, an offence which resulted in a 10-game ban, coming off the bench for 18 minutes during the team’s 2-0 friendly win over Melbourne Victory in front of a crowd of 95,000 at the Melbourne Cricket Ground.
Rodgers played down the latest offer from Arsenal post-match as he said: “There’s nothing new to report. He is very much a Liverpool player and over the course of the next couple of weeks we’ve got to get him up to speed.”
He added: “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.”
As well as the controversy over the Ivanovic incident, Suarez received an eight-game ban in December 2011 after being found guilty by an English Football Association hearing of racially abusing Manchester United’s Patrice Evra during a game.
He was supported by Liverpool throughout the case, with most notably his teammates and Kenny Dalglish, the manager at the time, wearing shirts with his name during the warm-up at the first game played after the verdict, at Wigan Athletic.
Suarez did little to dampen the speculation of his unhappiness at Liverpool as he cut a subdued figure during his cameo on the pitch, and did not join in the celebrations after setting up Iago Aspas’s goal in stoppage time, the second and clinching goal in the win over Melbourne.
Steven Gerrard had scored the opening goal in the 32nd minute after an exchange of passes with Joe Allen. The Liverpool captain said he and his teammates could take satisfaction from winning, even if it was not a competitive encounter.
“Every time you put the Liverpool shirt on you’ve got to give it your best and you want to win every game you play,” Gerrard said.
“But it doesn’t matter who scores the goals, it’s all about the team, and the team is in good preparation for hopefully a successful season. It’s another step forward.”
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