The 29-year-old, who moved to Stamford Bridge last January after an unhappy spell at Manchester United, has failed to make an appearance for The Blues as yet and manager Claudio Ranieri has admitted he would not be surprised if the player asked to leave.
Bosnich has found himself as third choice behind Ed de Goey and Carlo Cudicini at the West London club.
The Australian international had expected to be made first choice for the start of the season after an injury to Cudicini, only for Ranieri to start with de Goey, but he is determined to work his way into the side.
"I have signed for three years, and I am going to make it work," said Bosnich.
Ranieri had questioned Bosnich's fitness, but the former Villa shot-stopper insists when he gets fit he will prove his worth to the side.
"Everywhere I have been my contract has been my word. I know I have a battle on my hands to prove to the manager what I can do, but I am not here just for a short time.
"I know if I can stay fit, and just get a chance, I believe my future lies with Chelsea. It's the club I wanted to join in the place I want to live."
