The Gunners signed Germany international Bernd Leno in a £20m deal from Bundesliga giants Bayer Leverkusen last month.
Leno will be tasked with challenging Cech and Colombian goalkeeper David Ospina for a starting spot under Unai Emery.
Cech was handed the number one shirt following the conclusion of the 2017-18 Premier League season.
The 35-year-old has been the first-choice shot stopper since his £11m move to Chelsea FC from Arsenal in the 2015 summer transfer window.
Cech has been linked with a sensational return to Chelsea FC with Thibaut Courtois set to move to La Liga giants Real Madrid.
Former Arsenal midfielder Parlour has urged Cech to remain at the Gunners despite the temptation to secure a summer move.
“Cech won’t be looking to leave this summer and Arsenal will be counting on him being number 1,” Parlour told bWIN.
“I think [David] Ospina will leave the club this summer. rather than Cech.”
Arsenal finished in sixth place in the Premier League table last season to miss out on a spot in the Champions League.
The Gunners haven’t won the Premier League title since 2003-04, while their last Champions League final appearance came in 2005.
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