Lynagh, 48, is in a critical but stable condition at the Royal Brisbane and Women's Hospital (RBWH) Intensive Care Unit.
He was admitted on Monday night after complaining of blurred vision and headaches following a long-haul flight from London, where he now lives, to Brisbane.
Dr Rob Henderson from the RBWH said in a statement: "Michael has suffered a cerebellar and occipital lobe stroke due to a vertebral artery occlusion. This stroke has affected his vision, co-ordination and balance.
"This significant stroke is a rare event for someone of Michael's age.
"The next few days are critical because of swelling in the region of the brainstem, but current signs are positive."
Lynagh's father Ian said: "On behalf of Michael, I would like to thank the public and his friends for their well wishes and the overwhelming support for Michael.
"It has been a difficult few days and it is reassuring to know that there are so many people who care."
Lynagh won 72 Test caps and retired from international rugby in 1995 as the world record points scorer with 911, a total which remains an Australian record.
Inspirational playmaker Lynagh, who also won 100 caps for Queensland, made his Test debut in 1984 and was part of Australia's Grand Slam-winning team later that year.
Lynagh was vice-captain of Australia's World Cup-winning side in 1991 and scored the late winning try against Ireland in the quarter-finals.
After captaining Australia to the last eight of the 1995 World Cup, he retired from international rugby and joined Saracens at the start of the professional era.
Lynagh was the club's first major signing after Nigel Wray had taken control and he helped attract Philippe Sella, Francois Pienaar and Kyran Bracken to the club.
Lynagh spent two years at Saracens, guiding them to a 48-18 victory over Wasps in the 1998 Tetley Bitter Cup final and has been enrolled in the club's Hall of Fame.
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