The Liverpool striker, who was voted both the PFA and FWA Player of the Season, has been linked with a switch to one of the top clubs in Spain when the transfer window opens next month.
Suarez signed an improved deal with the Merseyside club in January after being the subject of a bid from Arsenal last summer - and after helping to guide Liverpool to second in the Premier League, Neville says he should publicly confirm that he wants to stay at Anfield.
"I think he's got a huge role to play in the next couple of months," he said on the Sky Sports End of Season Special.
"He's been the best player in the league this season, an incredible performer and he shows he cares because of the way in which he reacted in the last couple of weeks.
"I think he's got to come out and make a big statement. The rumours started the other day about Real Madrid and Barcelona and he's got to squash those straight away and that will give Brendan Rodgers the foundations to be able to mount a recruitment plan this summer.
"If there's nonsense hanging over him all summer it won't do Liverpool any favours. He's got to come out pretty quickly and put that to bed."
Newspaper reports have suggested that both Real Madrid and Barcelona are monitoring a potential €100million buy-out clause that may have been included in Suarez's latest contract.
And Neville says it may be time for Liverpool boss Rodgers to take a firm stand and insist his star player isn't for sale.
"Brendan Rodgers has to box him off now," he added.
"He can't allow him to go into a World Cup with uncertainty dragging on and uncertainty up in the air.
"I don't know what the clause relates to that we've seen in the last couple of days. I thought in January when he signed his contract that the clause related to not getting into the Champions League.
"If he hasn't got a clause that allows him to leave, I would just say 'you're not going' because at the end of the day Liverpool Football Club can't afford to lose him this summer or else they will struggle badly to replace him.
"I know Liverpool will survive and go on, but in terms of building on what they've achieved this year, Brendan Rodgers needs to get that dealt with quickly."
And Neville also paid tribute to the improvement he has seen from the player, both in terms of talent and attitude.
"He got to a level that I didn't think he was capable of when I first saw him two years ago," he added.
"I thought at that time his game lacked maturity and decision-making, but this year everything has come together at once. His appreciation of his team-mates is so much better, his finishing is absolutely fantastic and what impresses me most is his attitude towards the game. It's impeccable.
"Modern-day footballers get a lot of criticism because they don't fight, but every time you see him on a football pitch it looks as though his life depends on it.
"I have to say if you match that with the talent he has then you've got an incredible player."