The club's medical staff were informed of the test results as the team were preparing for Tuesday's League Cup third-round match against Hull City at London Stadium.
"All three, who are asymptomatic, will now follow Public Health England and Premier League guidelines and protocols," the club said in a statement, adding that the manager and both players immediately left the stadium and have returned home.
The match went on as planned with West Ham assistant Alan Irvine overseeing a 5-1 win against Hull.
The club statement added that West Ham had offered to conduct COVID-19 tests for the Hull City team before the match, but the visitors declined.
Regarding the decision to play on despite the positive tests, the EFL said in a statement: "Guidance permitted the match to go ahead as planned but not before the two clubs medical teams, in conjunction with the EFL's medical advisors, agreed all relevant protocols had been followed and both teams were comfortable in continuing."
Earlier in the day, Tottenham's Carabao Cup tie against Leyton Orient was postponed after the League Two side suffered a widespread coronavirus outbreak in their squad.