The former is re-joining the Gunners after impressing during a six-month loan spell in the 2020-21 season, while Ramsdale is set to compete with Bernd Leno for the right to wear the number one jersey.
Visa issues may prevent a roadblock when it comes to Odegaard's arrival, but Arsenal reportedly hope that both players will be available for their Premier League encounter with Chelsea on Sunday afternoon.
Here, Sports Mole looks at how the Gunners could line up against the Blues with Odegaard and Ramsdale in the side.
Arsenal are clearly not forking out close to £30m for Ramsdale to provide backup to Leno - the 23-year-old will eventually aim to become Arsenal's first-choice goalkeeper and will hope to quickly win over Mikel Arteta for a start against Chelsea.
Leno did not cover himself in glory for Sergi Canos's goal against Brentford and will have a fight on his hands to retain the gloves, while Pablo Mari and Ben White - who both looked shaky on the opening day - will continue in defence as Gabriel Magalhaes nurses a knee problem.
Thomas Partey's bad luck with injuries has ruled him out of the visit of the Blues, so the indispensable Granit Xhaka - who has reportedly agreed to sign a new deal at Arsenal - could be partnered by another new arrival in Albert Sambi Lokonga ahead of Mohamed Elneny.
Emile Smith Rowe may have been handed the number 10 shirt when he committed his future to Arsenal earlier in the summer, but during the second half of the 2020-21 season, Odegaard played in the hole while the Englishman attempted to weave his magic from the left-hand side.
Smith Rowe has made himself undroppable and could therefore find himself out wide against the Blues if Odegaard is registered in time, while Gooners will be praying that Bukayo Saka is fit enough to start after a cameo appearance against Brentford.
Uncertainty lingers over the availability of Pierre-Emerick Aubameyang and Alexandre Lacazette, so either Folarin Balogun or Gabriel Martinelli - the more likely option - seems set to lead the line against Chelsea, unless Arteta springs a surprise by fielding Nicolas Pepe up top.