Norgaard joined Arsenal from Brentford only last summer in a deal worth around £10m after six years as the Bees' captain, during which he made close to 200 appearances and established himself as one of the Premier League's most reliable defensive midfielders.
His first season in north London did not go to plan, however: he made just 20 appearances across all competitions, with only one Premier League start — in the final match of the season against Crystal Palace — as he found himself well down the pecking order under Mikel Arteta.
The 32-year-old is now poised to undergo a medical next week ahead of completing a move to Merseyside, becoming David Moyes' fourth signing of the summer following the arrivals of Hayden Hackney from Middlesbrough, plus Tyrique George and Merlin Rohl, both of whom spent last season on loan at Goodison Park. Norgaard is expected to fill the defensive-midfield void left by Idrissa Gueye, who departed as a free agent last month, giving Everton a ready-made replacement with a Premier League winner's medal despite his lack of game time.
Norgaard's departure clears the way for Arsenal to press on with a separate deal for Newcastle's Bruno Guimaraes, with the Brazilian's move to the Emirates reportedly closing in on completion following Arsenal's earlier agreed offer in the region of €90m (£77m).
Arsenal's willingness to let Norgaard leave for a fraction of what they paid for him a year ago reflects how quickly he fell out of Arteta's plans once other midfield options were established ahead of him.
A 41-times capped Denmark international, Norgaard will provide Moyes with experienced competition in the middle of the park alongside James Garner and the recently arrived Hackney, an area of the squad Everton were keen to strengthen following Gueye's exit.
Despite his limited minutes at Arsenal last season, Norgaard remained a consistent professional throughout, never publicly agitating for a move away even as his opportunities dried up, and leaves north London with a Premier League winner's medal to show for a season that otherwise offered him little on the pitch.