Mignolet has the challenge of displacing Jose Reina, the long-time first choice, even if the suspicion remains he has been brought in as a replacement for the Spain international.
Toure has the unenviable task of filling the boots of the retired Jamie Carragher, although if Martin Skrtel remains at the club there is a chance his centre-back partnership with Daniel Agger may stand in the way.
However, neither player will shy away from the challenge of upsetting the established order.
"He (Rodgers) spoke to me about that (the Reina situation) but it is not really something which will go outside of the club," said Mignolet, who will get his first chance to impress in Saturday's friendly at Preston with Reina still on extended leave after Confederations Cup duty.
"Competition is going to be in every single club. Everywhere you go, especially when you go to a bigger club, there's always going to be a fight for places and definitely when you're a goalkeeper because only one can play. But it's a challenge. It can only make you better and make you stronger in the end."
Toure's situation is slightly different in that he was released by Manchester City after a couple of years when he struggled to hold down a first-team place. He intends to make up for that frustration with a new lease of life at Anfield.
"I wanted to join Liverpool because of course I want to play more games," he said. "At City in the last two years, I haven't played a lot of games.
"Being there four years, two were very good and the last two years were much more difficult. The competition is tough, there are some good defenders here, we all need to fight for our place - I'm willing to do that."