The Gunners finished in fifth place last term under Unai Emery, 28 points behind champions Manchester City.
The club has not won the title since 2004, when the Invincibles claimed it.
Josh, who is the deputy chairman of Arsenal owners Kroenke Sports and Entertainment (KSE), told BBC Sports: “Our ambitions are the same as the fans.
“We want to win and we want to win as much and as often as possible. And doing it a fun way, where they’re seeing some really entertaining football as well. I think we’ve got the group to do it.
“We have the highest of ambitions. In North America, we are trying to win. The [Los Angeles] Rams were in the Super Bowl last year. I can only imagine what a Champions League final is like after being over in Baku.
“There are six great clubs in the Premier League and unfortunately only four [Champions League] spots that are guaranteed. The economics involved – to be able to reinvest back into the club, attract different players who only want Champions League football… our goal is to get back and to win the Premier League.”
