The 31-year-old Philippines international had recently reported back for pre-season training with the Blues.
Blues technical director Craig Gardner said: "On behalf of the board, the staff, the players and everyone connected to the club, we're all behind Neil and willing him on in his battle against Covid-19
"We're in constant contact with his close family and will continue to support them in any way we can.
"He's in the best possible hands and we're hopeful we can all welcome him home at the earliest opportunity."
Etheridge made 43 appearances last season after joining the club from Cardiff.
A tweet from the club read: "The club can confirm that [Etheridge] is currently receiving treatment in hospital for Covid-19. Everyone at #BCFC is sending him all our strength and best wishes."