Bernd Leno started in goal against Chelsea on Sunday, but Ramsdale is likely to come in for the second-round trip to The Hawthorns.
Odegaard could also start after getting visa clearance following his £30m move from Real Madrid last week.
Arsenal will still be without Ben White (Covid-19), Eddie Nketiah (ankle) and Thomas Partey (ankle), but Gabriel could make the squad as he recovers from a knee issue.
Alexandre Lacazette could also make the bench after recovering from Covid-19, while Kieran Tierney and Gabriel Martinelli have shrugged off cramp issues from the weekend.
Arsenal likely XI (4-2-3-1): Ramsdale; Chambers, Holding, Mari, Tavares; Elneny, Xhaka; Saka, Odegaard, Smith Rowe; Aubameyang
Tonight 8.00pm BST, Sky Sports
Arsenal squad from: Leno, Ramsdale, Runarsson, Okonkwo; Bellerin, Cedric, Chambers, Gabriel, Holding, Kolasinac, Mari, Tavares, Tierney; Elneny, Lokonga, Maitland-Niles, Odegaard, Saka, Smith Rowe, Torreira, Xhaka; Aubameyang, Balogun, Lacazette, Martinelli, Nelson, Pepe and Willian
Doubts: Bellerin (thigh), Gabriel (knee), Willian (Covid-19)
Out: Nketiah (ankle), Partey (ankle), White (Covid-19)