The West Yorkshire club have tied up five out-of-contract players, including most recently Danny Brough, since the start of the season but Australian Stone says he has yet to begin talks over his own position.
He is out of contract at the end of this year having replaced Paul Anderson in July 2016.
Stone, 51, helped Giants avoid relegation in his first campaign and then guided them to a top-eight finish last term.
He said: “Ideally, I’d love to stay here and build some success that’s sustainable and keep the club in good health, making sure particularly some of the young boys coming through, blossom.
“That would be great. It’s really important for a club (like) ours, who are not quite in the market as some of the bigger hitters to be able to produce our own talent.”
Stone insists he is not concerned over the uncertainty as he prepares his side for Thursday’s televised game against Hull KR at the John Smith’s Stadium.
He added: “My immediate focus is winning some games.
“Ideally everyone would like to know what their situation is but it doesn’t make any difference to how you prepare or what the results are.
“Maybe I’ll sit down and talk to (chairman) Ken (Davy) and (managing director) Richard (Thewlis) in due course and find out what the future’s going to hold.
“Maybe they’re just waiting to see what happens as well themselves.”
Toronto Wolfpack have stepped up their push for a place in Super League with the capture of Gareth O’Brien from Salford.
The Canadian outfit are thought to have paid a fee of around £100,000 for the 26-year-old full-back, who signed a new three-year contract with the Red Devils just under 12 months ago.
Tigers forward Oliver Holmes will appear before the Rugby Football League disciplinary committee today charged with a grade D dangerous throw in Sunday’s win over Salford.
Holmes was sin-binned for the tackle on Red Devils’ Tyrone McCarthy.