A Foxes spokesperson told the Bristol Evening Post: "We felt we made a very favourable offer for Nicky Maynard and were surprised when it was turned down.
"Bristol City have made their position clear and it is time to move on and look elsewhere."
A new contract has been on offer for Maynard, but he has yet to reach an agreement and Bristol City manager Keith Millen is concerned Leicester's interest may have turned the player's head.
"I'm worried that all this speculation could turn out to be a distraction for Nicky," said Millen.
"It is part and parcel of the game these days, but it is not ideal for the club or the player. I don't want Nicky's head being turned by other clubs, I want him to be fully focused on getting fit and match sharp, so that he is at his best when the season kicks off in August.
"I'm ambitious and want us to be pushing for promotion next season. I'm certainly not in the business of selling my best players and I'm delighted the club stood firm and rejected the bids from Leicester.
"Nicky is happy here and, apart from anything else, the money on offer was nowhere near enough."