Arsenal have seen two bids for the Uruguayan rejected - the latest being £40,000,001 in an attempt to trigger a supposed release clause - only for Arsene Wenger to recently admit defeat in the pursuit.
And Gerrard is confident Suarez will now stay at Anfield for at least another season and he does not class any potential move to the Emirates Stadium as a step forward for the striker.
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"I was frustrated too when Luis was just being linked with Arsenal. I don't want to disrespect Arsenal in any way. They are a fantastic club but I don't see it as a step forward for Luis in his career. There are only two or three clubs he should leave this club for if he is taking a step forward.
"I understand we're a little bit off the Champions League at the moment but it would be a shame to ruin the relationship he has got with the supporters to go to another English club. I saw what happened to Fernando Torres, I don't think he should make that type of move and that's the advice I gave to him."
Gerrard also thanked the Liverpool fans for continuing to back Suarez despite the former Ajax man making no secret of his desire to leave the club, and now hopes he will show some recognition to the supporters.
"The support the fans have given him has been phenomenal," he says. "They haven't once jeered him or tried to push him or fall out with him. They absolutely idolise him, so surely he wants to walk out with his head held high and go to a Real Madrid, Barcelona, Bayern Munich.
"He can walk into any side in the world, he's that good. No one would begrudge him such a move, but for me it is important to be clapped out and clapped back in if you ever return in the future.
"People talk about player power but players have got to realise there is a loyalty they need to pay back at certain times. Luis has got to openly admit that the club has stood behind him on a couple of massive occasions and rewarded him with new contracts."