The BBC reported that the Uruguayan wants to speak with the Gunners after they communicated a bid of £40 million plus £1 to the Reds, which American owner John W Henry laughed off on Twitter.
The Gunners made the unusual offer because under the terms of the deal that Suarez signed in August last year, any bid over £40m would compel
"There's no doubt the market these days for top players is very small so Luis will always be linked with top clubs," Rodgers said after Suarez made a cameo appearance in Liverpool's 2-0 win over local club Melbourne Victory on Wednesday.
"But I think the support that he's received from the supporters and the people of the city of Liverpool has been unrivalled.
"In the period of time, he's missed a lot of games for the club through various reasons. And the people have stood by him like a son and really looked after him.
"So I'm sure whatever happens in the coming weeks, that will be in his mind because it's certainly something you can never forget."
Suarez was due to discuss his future with Rodgers in Melbourne after rampant speculation about his future went up a notch following reports Arsenal had made a second bid for the 26-year-old.
Suarez has said he wants to leave Liverpool to play in Spain, blaming the British media for making his life impossible, with Real Madrid thought to be interested but yet to lodge a bid.
Rodgers, who left Suarez on the bench until the last 20 minutes in front of more than 95,000 fans at the Melbourne Cricket Ground – he then set up Iago Aspas for Liverpool's second with a trademark piece of trickery - said he had already shared several chats with his tempestuous striker since he landed in Melbourne.
Suarez, who signed for Liverpool from Ajax for£22.7m in January 2011 and was second top scorer in the Premier League with 23 goals last season, has tempered his brilliance on the pitch with a series of controversial misdemeanours.
He still has to serve six matches of a 10-match ban for biting Chelsea's Branislav Ivanovic in April, which excludes pre-season friendlies.
Suarez also sparked a storm of criticism the previous season when he was found guilty of racially abusing Manchester United's Patrice Evra and earned an eight-game suspension.
Rodgers had "nothing to report" on reports of the latest move by Arsenal, who have already had one bid for Suarez rejected by Liverpool.
"The only talks have been between me and Luis," he said.
"He's a little bit tired at the moment. He's trained a double session on Monday after being off for three weeks so he was a little bit tired and, obviously, he came on and played that 20-odd minutes or so."
Suarez was among the last players to leave the MCG turf, saluting cheering fans in a victory lap, some of whom may wonder whether it was a valedictory bow in Liverpool colours.
"He's very much a Liverpool player," Rodgers added. "Over the course of the next number of weeks we've got to get him up to speed."
Rodgers said last week that he considers him to be worth at least as much as Edinson Cavani, who was sold by Napoli to Paris St-Germain for £55m.
- Sports & Recreation