Rodgers was appointed Leicester manager earlier this week, leaving Celtic after two-and-a-half years in Glasgow.
The former Liverpool boss is now the third permanent manager at the King Power since the sacking of Premier League-winning Claudio Ranieri in 2017.
Numerous reports in the last couple of years have claimed the more experienced players at Leicester have had an influence on the future of managers.
Peter Schmeichel, father of Foxes keeper Kasper, confirmed earlier this year his son was not totally happy at the club.
Star striker Jamie Vardy also had friction with previous manager Claude Puel, being left out of the team on a number of occasions.
Souness was asked by Sky Sports presenter David Jones what Rodgers should do with the likes of Schmeichel and Vardy.
“What would you do with the big characters there? The likes of Kasper Schmeichel, Jamie Vardy,” Jones asked.
Souness says it is clear the duo, among others, have had an impact on the managerial merry-go-round at Leicester.
And he believes it is key Rodgers gets them on side early in his reign or he will feel “vulnerable”.
“Well I think it’s obvious that they have an influence when things are not going well,” Souness said.
“They’re not slow to voice an opinion, whether it’s directly or through a family member or an agent, which I don’t like.
“I didn’t see that group being supportive for larger part of Puel’s time at that club.
“I think there’s no doubt about it that those players have access to the highest member of that football club, I mean the owner.
“He has to walk in there and get those big players on side.
“There are issues at that club, I think issues maybe at lots of clubs, but if the players can bypass the manager so regularly as it would appear as they do at that football club, then the future isn’t great because the manager is always going to feel vulnerable.
“If you’ve got a few disappointing characters in your dressing room, who feel they can undermine the manager when things aren’t going well, they will do it.
“I look at the stories that come out of Leicester, not recent stories but going back, I come back to it and the reason I say that and I say it with great confidence is that that’s a team that won the league three years ago and the following year they were fighting relegation near enough.
“What does that say about that group of players? That means none of them held the mirror up and said ‘could I do more?’, they were looking for someone to blame.
“They’ve got to get those senior ones on side. We’ve all been in dressing rooms, we know the senior players, some can be all important, they set the standards and they set the influence throughout that dressing room.
“He has to get them on side in the early days.”