The Englishman made 133 appearances for the Reds during a six-and-a-half-year stint at Anfield that began when he arrived as an 18-year-old in 1951.
"The thoughts of everyone at LFC are with Brian’s family and friends at this sad time," Liverpool said in a statement.
Signed from Leyton Orient by Don Welsh, Jackson marked his debut for his new club by scoring against Bolton Wanderers.
The Surrey-born outside-right would go on to net a further 10 goals for Liverpool during a period that saw them relegated to the Second Division and struggle to bounce back.
We are deeply saddened by the passing of former player Brian Jackson, aged 86.
The thoughts of everyone at the club are with Brian’s family and friends at this sad time.
RIP Brian 1933-2020— Liverpool FC (@LFC)February 18, 2020
He left for Port Vale in the summer of 1958, just over a year before Bill Shankly arrived to revolutionise the Anfield club.
Jackson also went onto play for Peterborough United, Lincoln City, Burton Albion and Boston United before calling time on his career.