Odegaard will wear the No8 shirt, but he is unable to make his debut this weekend against Chelsea as he is still awaiting visa clearance.
Odegaard arrived at the Emirates in January last year and impressed during the second half of the campaign, with Mikel Arteta stating at the end of the season that he was hopeful the Norwegian would return.
Arsenal have forked out around £30m to sign Odegaard, who is keen to link up with Mikel Arteta again and return to north London.
At one stage this summer that looked unlikely, but the Gunners redoubled their efforts to sign the playmaker after the sale of Joe Willock to Newcastle last week and he has now been officially unveiled as an Arsenal player for the second time.
The news is a considerable boost to Arteta, who is already under pressure after his side lost their opening game of the Premier League season away to newly promoted Brentford on Friday night.
Arsenal’s creative woes were on show once again, with Odegaard expected to provide competition and support for academy graduate Emile Smith Rowe, who has been handed the No10 shirt for the new season.