The former Arsenal and Swansea goalkeeper, who joined West Ham in 2018, made his 300th Premier League appearance in the 2-0 win over Norwich which lifted the Hammers back into the top four.
Fabianski, 36, whose current deal expires in the summer, is expected to be offered a new one-year contract.
He said: “I hope so. I don’t know what to say, it is always a tricky question.
“I am very happy in this present time and present moment. I am really focused on doing well for the club and if everything goes well then I don’t see any other options.
“It is a really special time to be part of this club at this moment in time. The club is going in a really good direction.
“We’re going really well in the league and managed to qualify for Europe so that is something special and I am really happy to be part of it.
“It is different. When I left Arsenal the main thing was to play regularly as a number one and there was always a dream to be with a club that would fight for something more, and I think West Ham is in a moment that they are trying to take another step forward and be a club that will compete for European places.”
Jarrod Bowed scored both goals against Norwich to enhance his England credentials and Fabianski believes the winger deserves a call-up.
“I think he has been amazing since he signed for the club, from the very first moment” added Fabianski.
“He is so direct but I think what he has improved is his finishing, the quality of his runs and the timing of his runs.
“The final bits as well are there. He is strong and powerful, we are happy to have him.
“Any defender that plays against him struggles a bit so he has become a big threat and in my opinion he really deserves a chance with the national team as well.”
A sixth straight defeat, all without scoring, left Norwich rooted to the foot of the table but by no means cut adrift.
“We’re still only three points from safety and I have got a committed and honest bunch of players,” said manager Dean Smith.
Todd Cantwell missed out through illness but Smith expects the midfielder to be available to face Everton on Saturday along with Josh Sargent, who was absent after his wife gave birth.