The former Reds player and manager was embarrassed after the Uruguayan appeared to bite the Chelsea defender before bagging the equaliser in the 2-2 draw on Sunday
Suarez, who equalised for Liverpool in added time to earn a point against Rafael Benitez's team, drew the ire of Ivanovic in the second half of their Premier League clash on Sunday afternoon.
The former Ajax striker has previously been banned for racial abuse and gesturing towards opposition supporters and Souness believes the latest controversy could put his future at the club in doubt.
"I'm not sure what to make of it," he told Sky Sports. "Embarrassing. He's making it very difficult for himself to stay at Liverpool, that puts him in the last-chance saloon.
"Most important is safeguarding the good name of the football club. Liverpool are world renowned. People will be talking about this for a long, long time, and it will show Liverpool in a very bad light.
"This is not the first time he has bitten someone in the football match [Suarez was banned for seven matches for the same offence while playing for Ajax in 2010].
"He is risking everything this great club stands for. I've never seen something like that in a football match before. There are lines you just can't cross, and he has crossed it today.
"Those pictures will be beamed across the world. It's very difficult for Brendan Rodgers to defend him. We will get politicians involved in this and everything."