“I’ve just turned 31 and had signed a new deal with Brentford,” he said.
“There’s always a little chance that a miracle could happen with Brentford, but it was probably unrealistic to play Champions League with them.
“Now I get to hear the anthem and be a part of a Champions League campaign. It’s a childhood dream and something I didn’t think would be possible. So it’s something I’m really, really happy about.”
Norgaard helped Brentford win promotion from the Championship in 2021, before making 122 Premier League appearances.
He is Arsenal’s third signing of the summer.
Midfielder Martin Zubimendi joined from Real Sociedad last week in a deal worth almost £60m, while goalkeeper Kepa Arrizabalaga moved from Chelsea for £5m.
Arsenal are in talks to sign Sporting striker Viktor Gyokeres for £70m and have made a formal approach for Chelsea winger Noni Madueke.
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