Lampard discussed the winger's situation after the squad touched down in Tokyo for the latest leg of their preseason tour.
He said, "With Callum of course I want him to sign and stay here. He's a player who's come through the Academy; he's a player who can be a big player for Chelsea and England. We as a club as a whole want him to stay, for sure.
"I think he knows he has a manager here who wants to work with him to improve, and that takes hard work. Nothing is given, especially when you're 18 years of age, but with all the attributes Callum has we want him to come through at Chelsea and be a big player for us now and in the future.
"There a few younger players now, some have signed, some I think will sign in the next month or in the early part of the season, and it's important we move forward on that. I know as a club we are doing our best from top to bottom."