Bernd Leno started Sunday in goal against Chelsea, but Ramsdale is likely to reach the second round at The Hawthorns.
Odegaard could also start after obtaining a visa permit after his move of £ 30 million from Real Madrid last week.
Arsenal will continue without Ben White (Covid-19), Eddie Nketiah (ankle) and Thomas Partey (ankle), but Gabriel could form the team when he recovers from a knee problem.
Alexandre Lacazette could also take the bench after recovering from the Covid-19, while Kieran Tierney and Gabriel Martinelli have ruled out ramp problems since the weekend.
Arsenal probably XI (4-2-3-1): Ramsdale; Chambers, Holding, Husband, Tavares; Elneny, Xhaka; Saka, Odegaard, Smith Rowe; Aubameyang
Tonight at 8pm BST, Sky Sports
Arsenal selection of: 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)
Outside: Nketiah (ankle), Partey (ankle), white (Covid-19)