The 28-year-old is in a stable condition and in communication with his family but will be unavailable for selection while he continues to be monitored by doctors.
Wigan chief executive, Mal Brannigan, said: "First and foremost, the absolute priority for everyone associated with Wigan Athletic is Charlie's health and wellbeing and providing support to him and his family.
"As a club, we would like to place on record our sincere appreciation to all of the staff within the NHS who, within what we know are challenging times, were excellent in their response and subsequent care for Charlie, working together with our medical team.
"Clearly this news will cause alarm, though we would politely request that the privacy of Charlie and his family is respected. At the appropriate time, we will look to provide further updates to our supporters."
The news comes two days after Sheffield United midfielder John Fleck collapsed during the Blades' clash at Reading. Fleck was discharged from hospital the next day.
Wyke joined Wigan this summer from Sunderland and has scored five goals in 17 appearances for the Latics.
Club owner Talal Al Hammad wrote on Twitter: "I'm praying that you get better and regain your strength soon Charlie. You are in all of our wishes as you recover. We are all keeping you in our thoughts and prayers, hoping you find strength with each new day. Get well soon."
Many others have sent Wyke well-wishes, including former clubs Sunderland and Bradford.
Quote tweeting Wigan's social media post announcing the news on Wyke, Sunderland posted with two heart emojis: "Get well soon, Charlie."
Bradford tweeted with two heart emojis: "Sending our very best wishes, Charlie."
Meanwhile, the EFL said with a heart emoji: "Get well soon, Charlie Wyke."