Germany international Leno arrived at the Emirates from Bayer Leverkusen in a deal worth £19 million ($25m) this summer as one of new coach Emery's first signings at the club.
The 26-year-old is expected to replace veteran Cech as the Gunners' first-choice shot-stopper in the coming season, while it has been suggested that the 36-year-old may look to leave the club after three years of service.
Cech has moved to quash such reports, however, insisting that his only intention is to prove his worth to his new manager.
"I have a contract with Arsenal," Cech told the official Arsenal site. "I am concentrating on this season, on the competition between the goalkeepers because we have plenty of options.
"I'm doing my best to show to the manager that I can be in the starting lineup when the season starts.
"That's my aim and as for the speculation, I don't really follow it. I've had no contact with Chelsea."
Cech has also backed Arsenal team-mate Mesut Ozil following his retirement from international football with Germany, for which he cites "racism and disrespect" as the reason for his decision.
Ozil has received a backlash from certain areas of the German media but Cech is hoping that his decision to focus on club football will benefit the Gunners in the 2018-19 Premier League campaign.
"I have to say Mesut has come back and is really happy to be here. He's training really well," Cech said.
"He's happy to be with us and we're happy to have him back as well because he is one of the key players in our team and we need him to be fit and ready for the season. He has been working really hard so hopefully he will be fit.
"As I said Mesut is an important player for us, we want him to perform, we want him to feel comfortable. He has the support from the team and we hope by the start of the season he will be fit and ready to perform for us."