The Poland international was dropped for the Champions League game against Bayern Munich last week as manager Arsene Wenger felt the 22-year-old had been "mentally affected" by the number of games played, and again took him out of the firing line for Saturday's trip to Swansea where Lukasz Fabianski kept a second consecutive clean sheet.
Despite maintaining he could offer no players a guarantee of a place in his side - not even captain Thomas Vermaelen, who also sat out the last two Gunners matches - Wenger insists Szczesny remains part of his long-term plans, having taken a chance on the young Pole who came through Arsenal's youth ranks after joining the club's academy from hometown side Legia Warsaw in 2006.
Speaking to Poland daily newspaper Gazeta Wyborcza in an interview conducted before the Swansea game, Szczesny said: "I am too happy to go somewhere else. I believe in the philosophy of the club and that Arsenal will start to win again.
"I have a debt to pay here for the club, I owe them. I just do not want to go anywhere else.
"As long as they want me here, there is no 'somewhere else', and I have a contract until June 2015."
Arsenal beat Swansea 2-0 away on Saturday as they battled back into contention for a top-four finish, with Tottenham's 1-0 home defeat by Fulham leaving them four points behind their local rivals, with a match in hand and a superior goal difference.
Szczesny feels it is unthinkable for Wenger's men to miss out on playing in the Champions League next season.
"It is halfway through March and we have lost everything. There are still 10 matches left to save the club against the humiliation of all times.
"It would be disaster not to be in the top four and without Champions League next year.
"We have the 10 most important games in our careers. We are going through the most difficult time since I came here."
Defender Carl Jenkinson feels Arsenal need to focus on delivering consistent performances from their own remaining nine matches, following the win at Swansea, rather than worry about the shortcomings of others.
Speaking to Arsenal Player, Jenkinson said: "We just need to concentrate on ourselves. Other teams are going to drop points.
"We want to win every game and we are capable of doing that.
"Our form hasn't been bad. It can be better of course. From now it's just about now until the end of the season. That's going to be what makes or breaks us, so it's almost like a little season in itself.
"We just need to get on a good run and keep it going. If we do that, we'll end up where we need to be."
Arsenal beat Swansea without England midfielder Jack Wilshere, who is nursing an ankle problem independent of the injury which sidelined him for more than a year.
Reports on Monday afternoon suggested the 21-year-old, who has been allowed time away in Dubai to aid his rehabilitation, is set for an extended spell on the sidelines.
However, Wenger expects to see him return after the international break, with Arsenal's next Premier League game against Reading on March 30.