The Norway international, who will join for an initial £30m fee plus bonuses, was in London on Wednesday undergoing a medical.
Personal terms are not expected to be a problem and Sky Sports News has learnt Odegaard is fully vaccinated, so will not have to quarantine.
If he is registered by midday on Friday, Odegaard will be available to face Chelsea on Sunday.
Norway international Martin Odegaard made 20 appearances for Arsenal during the second half of last season; Gunners manager Mikel Arteta said he wanted more additions to his squad before transfer window closes on August 31
The 22-year-old spent the second half of last season on loan with the Gunners after failing to break into the Real Madrid team under Zinedine Zidane.