Oliver Skipp also resumed training with the squad on Monday but Antonio Conte may opt for more caution with the midfielder, who has not played competitively since January following a niggly pelvis injury last term, and a fractured heel sustained during pre-season.
Spurs on Tuesday announced they had triggered their option to make Romero's loan from Atalanta into a permanent move worth £46.5million, with the Argentine agreeing a five-year deal to 2027.
Romero has made 32 appearances for Spurs and established himself as a vital part of Conte's back three.
If he is deemed fit enough to face the Hammers, Davinson Sanchez is likely to make way, although Conte will be conscious that Spurs face a quick turnaround to Saturday's home game against Fulham and has promised to rotate his squad in the next three weeks.
Asked about his team selection on Monday, he said: “About Romero and Skippy. Today they did the training session with us. For sure we have to wait, to tell that [if] they are completely recovered, but this is a good sign for us, for the players.”
Lucas Moura and Bryan Gil will also miss Wednesday's game, while Rodrigo Bentancur is being assessed for concussion after coming off with a head injury in Sunday's 2-0 win at Nottingham Forest.
If he is deemed to have suffered a head injury, the Uruguayan would have to sit out for six days, ruling him out of West Ham and Fulham.
Conte added: “About Lucas, he has this inflammation in his tendon. I hope to recover him very soon, because for sure when I was a player I had a lot of pain in my tendon.
“You know very well tendon inflammation doesn’t finish in only one week, two weeks. I hope to have him early with us.
“Gil is not still available but in two, three days he will start to work and to recover.”