Tuilagi, whose current deal runs out next summer, has been in talks about extending his stay since September.
The 24-year-old is permitted to talk to other clubs in January.
"We have put our best foot forward contractually. He wants to stay," Cockerill told BBC Radio Leicester.
"We've made an offer that would put him up there with the highest-paid players in the world and we are waiting for that answer."
Tuilagi, who came through the academy system at Welford Road and made his first-team debut in 2009, has won 25 caps for England.
Cockerill said it is understandable there would be interest from other clubs in one of Leicester's biggest assets.
"If players want to look around they'll look around and I understand that," said Cockerill.
"That's part of getting your market value and finding your price in the market place.
"It's always flattering when you get courted by other people and we just have to be patient.
"We would like to think we would get that done sooner rather than later.
Tuilagi is sidelined with a groin injury that will keep him out until 2016.
"He's progressing really well," said Cockerill. "I don't really want to put a deadline or comeback date on it.
"When he is back he is back, but hopefully it is early in the new year.
"There's another 10 years left in Manu and, certainly in the next three years, we'd like him to be here and we have made that known."
Leicester director of rugby Richard Cockerill is confident England centre Manu Tuilagi will sign a new contract which will make him one the "highest-paid players in the world".