Suarez had claimed that he had an agreement that he would be allowed to leave Liverpool if they failed to qualify for the Champions League.
Arsenal have made two bids for the striker, and in the meantime Liverpool boss Brendan Rodgers has ordered Suarez to train alone and only return after he has apologised to his team-mates.
But his desire to leave now appears to have changed.
Suarez is currently away with Uruguay for their friendly in Japan and he was quoted in Uruguayan newspaper El Observador as saying he is willing to remain at Anfield because of the supporters.
"For now, owing to all the affection of the people, I would be staying," he said.
The newspaper goes on to suggest he may even extend his contract with the club.
That would represent a remarkable U-turn from Suarez.
In interviews with the Guardian and the Daily Telegraph, Suarez said: "Last year I had the opportunity to move to a big European club and I stayed on the understanding that if we failed to qualify for the Champions League the following season I'd be allowed to go.
"I gave absolutely everything last season but it was not enough to give us a top-four finish - now all I want is for Liverpool to honour our agreement.
"I don't feel betrayed (by Liverpool) but the club promised me something a year ago just as I promised them that I would stay and try everything possible to get us into the Champions League.
"They gave me their word a year ago and now I want them to honour that. And it is not just something verbal with the coach but something that is written in the contract. I'm not going to another club to hurt Liverpool."
He even threatened to take Liverpool to Premier League arbitration to secure his release.
Suarez said last week: "I have the club's word and we have the written contract and we are happy to take this to the Premier League for them to decide the case but I do not want it to come to that. We have the backing of the PFA."
But his latest comments suggest that the striker could well be staying which would be a huge boost for the club ahead of the new season.
Speaking on Tuesday, Liverpool captain Steven Gerrard admits he is desperate for his Anfield team-mate to stay on Merseyside.
"If I can use my influence to try to make him stay, I will try to do that," he told a press conference ahead of England's friendly against Scotland. "For Liverpool to move forward, it's important we keep our best players.
"I really don't know what is going to happen with him but I'm really confident he won't be going to anyone else in England."
Earlier in the week, Liverpool boss Brendan Rodgers stated there can be "no solution" to the stand-off with Suarez as long as the striker refuses to apologise.
He told the club website: "It is something that we have to do everything we can to fix.
"There has been a lot said and a lot reported, and we have made a stance, as a club and as a manager, of the commitment and the standards required.
"If you don't have that commitment, and those standards, then you won't play. It's as simple as that.
"If we have got to sit and wait, then so be it. But this is a club that has got great values, and we hope that the situation can be resolved amicably, and I am sure it will be.
"There's no problem between me and him, it is quite calm. But obviously I have got to respect the club and the team.
"It doesn't matter who you are, if they're not pulling their weight then they are letting down the club, the city and everyone who has stood by them.
"So until I get that assurance that we are going to get nothing but 100% commitment, then there will be no solution."
Reds defender Kolo Toure welcomed Suarez's apparent change of heart, telling Radio City: "It's great, we are all really happy.
"We want him to stay because what he did last season for the team was amazing. He's a quality player. We need to keep all of our quality players."