Houllier led Liverpool to a treble in 2000/01, with Gerrard captaining the side to the League Cup, FA Cup and UEFA Cup.
"The thoughts of everyone at Liverpool Football Club are with Gerard's family and many friends", the club said in a statement.
"Absolutely devastated and heartbroken at the sad news of the passing of Gerard", Thompson wrote.
A fan favourite at Anfield, Houllier suffered a life-threatening vascular problem during a Premier League match against Leeds in 2001.
French President Emmanuel Macron also paid tribute to Houllier, along with many current and former Liverpool players.
A former amateur player turned English teacher, Houllier started his coaching career with second-division French team Noeux-les-Mines before joining Lens.
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In 1985, Houllier was appointed PSG boss and led them to the top-flight title in the 1985-86 season.
However, he resigned in November 1993 after France failed to qualify for the 1994 FIFA World Cup finals.
After working with France's Under-18 and Under-20 teams, Houllier was named as co-manager alongside Roy Evans at Liverpool - a city where he had once worked as an assistant in a school.
France have qualified for every World Cup since then, lifting the trophy in 1998 and 2018.
During his career Houllier won the Ligue 1 title with Paris Saint-Germain in 1986, later winning it twice more with Lyon.
Five of those honours were secured in a memorable 2001, with the Reds enjoying a remarkable run on the trophy front.