Suarez has been banned from all football activity for four months for his third biting offence, an attack on Italy's Giorgio Chiellini while playing for Uruguay at the World Cup.
The striker has been heavily linked with Real Madrid and Barcelona in recent weeks, and FIFA's ban does not preclude a transfer this summer.
But Aldridge, himself a former Anfield centre-forward, told Sky Sports News the length of Suarez's ban will scare off potential buyers.
"I think for your Barcelonas and Real Madrids it's untenable now to come and take him knowing he's got a worldwide ban for so long," he said.
"I think the only possible situation now is for him to stay at Liverpool, and perhaps in the long-term stabilise his position as a Liverpool player.
"Everyone's got to buckle down and make sure nothing like that ever happens again. Three times is bad enough, but if it were to happen again that would be it.
"There may be some people want to turn their back on him, but I think the club will try and stick by him. In a Liverpool shirt he's been impeccable. they just have to work with him, work on whatever triggers it off and make sure it never happens again."
Suarez was banned for 10 club matches for biting Chelsea's Branislav Ivanovic last year, but the suspension did not affect Uruguay.
Aldridge said: "FIFA should perhaps have looked at the bigger picture: why should Liveprool suffer? Ban him from international football for the rest of his life - that would hurt him just as much, we know how passionate he is about Uruguay. If he never played for them again, would that make a difference?
"It is harsh if you think last time Uruguay got away with it and he was allowed to go and play for them, so I think there'll be a lot of Liverpool fans not too happy. But we all know it can't be condoned.
"FIFA has got to get the message over to children as well as adults that it just simply can't happen."