The Colombian goalkeeper is set to challenge Szczesny for the No 1 jersey this season.
Ospina signed for the Gunners from Nice this summer for a fee of around £3million, and is expected to challenge Szczesny for the number one jersey.
The Colombian was in top form for his country at the 2014 World Cup in Brazil, and his performances impressed Arsenal manager Arsene Wenger, who was in the market for a shot stopper to replace Lukasz Fabianski.
Szczesny, who has been Wenger's first-choice goalkeeper for the past few seasons, says he expected a new arrival and is looking forward to competing with Ospina.
"I love a challenge," Szczesny said, speaking to the Arsenal website. "I was expecting someone to come because Lukasz has left for Swansea and I wish him all the best. I knew someone would come in.
"I'm just going to do the same thing over again, I'm going to try my best in every game and every week in training. Hopefully that will be enough."
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Since being promoted to the Arsenal first team in 2010 Szczensy has had a firm grip on the number one jersey at the Emirates. He was briefly dropped in March 2013 after a string of poor performances, but rediscovered his form in a stellar 2013-14 campaign.
In signing Ospina the Gunners follow the trend of good teams having two top-class goalkeepers at their disposal. Szczesny will be hoping that his performances next season will ensure that Ospina remains one of the Premier League's top backups.