The 60-year-old believes the player owes Liverpool for the support they have given him during his controversies since he joined in January 2011
Suarez has been the subject of two bids from rivals Arsenal over the summer but owner John W. Henry reiterated the stance that the Uruguay striker is not for sale on Friday.
The 26-year-old claims he was promised he could depart for a Champions League club but Souness believes he should show loyalty after the club defended him during incidents that put the club's reputation into disrepute.
"I think he does [owe Liverpool] and everyone in their correct mind would see it like that," Souness told Sky Sports News.
"I think back to the t-shirts at Wigan when he got himself in real bother and the [Branislav] Ivanovic situation last season - the board have stood by him, the club have stood by him and the manager has stood by him.
"You would like to think a bit of common sense and loyalty would prevail but that doesn't appear to be the case."
Souness agrees with the sentiments of Professional Footballers’ Association chief executive Gordon Taylor that the sale of the player is near impossible due to the task of replacing him.
And the former midfielder - whose seven-year spell at Anfield included five league championships, three European Cups and four League Cups - now believes the club should tell him to stay for another season.
He continued: "If Henry really believes what he is saying [that Suarez will not be sold] then maybe he could say to him: 'Give me another year,' like Sir Alex Ferguson did with Cristiano Ronaldo.
"It is all too late for Liverpool now as they would have to pay a premium on players to replace him and get taken to the cleaners - and there are no obvious replacements."