Rodgers' work with the Foxes has him being linked with bigger managerial jobs.
Elliott told Goal: "Would I go to Tottenham if I were Brendan? Honestly, no. He has done a fantastic job in Leicester and it is a club with great potential.
"He has been rumoured to be interesting other big clubs as well and I understand why he is linked with those clubs.
"But he should stay in Leicester, I really think so. But then it does not always turn out as one hopes."