The American, who originally joined as a mechanic before rising through the ranks, was taken unwell last week.
A McLaren spokesman said: "We can confirm that Tyler has been taken ill.
"He was admitted to hospital last week, is undergoing tests, and is receiving the best possible care."
McLaren expressed its hope that he makes a quick recovery.
The team tweeted: "Best wishes to McLaren legend Tyler Alexander, who's in hospital at the moment. #GetWellSoonBuddy"