The 63-year-old Villa icon died after suffering a "severe head injury" when he fell from a taxi following a night out with friends in September.
The 1982 European Cup winner was regarded as one of the club's best forwards in the club's 150-year history, scoring 79 times in 213 appearances.
Before he passed away, Shaw had asked a pal to sell his historic medals if he ever died - ideally to the football club or a Villa fan.
They included his 1982 European Cup Winners medal, 1982 European Super Cup Winners medal, 1981 Football League Championship medal , 1982 Bravo Award medal and 1981 PFA Young Player of the Year Trophy.
A rare and valuable collection of football memorabilia belonging to the late Gary Shaw, former English professional football striker for Aston Villa,
The haul was expected to fetch between £50,000 to £70,000 when it went under the hammer at Mullock Jones Auctioneers in Shropshire last week (Thurs).
However, the collection failed to meet its reserve price with the highest bid reaching £39,500.
Before the sale, auctioneers said it was a "rare opportunity to own a true piece of football history".
Former Villa player Stan Collymore even took to social media to urge the club to buy the medals.
He wrote on X: "No better starting point to create a club museum worthy of our history than to buy a club legend's medals back, for future Villans to know the name of Gary Shaw, and the team he played with.
"Those medals deserve to be at Villa Park, B6."
A rare and valuable collection of football memorabilia belonging to the late Gary Shaw, former English professional football striker for Aston Villa,
The auctioneers said Gary offered a close friend the chance to purchase the medals around six years ago as the family were big Villa fans.
An agreement was made that the medals and trophy would never be sold while Gary was alive but to be put up for sale after he had passed.
Speaking before the sale, Ben Jones, from Mullock Jones Auctioneers, said "Gary Shaw was an integral part of Aston Villa's history and a cherished figure in English football.
"The items being sold reflect a truly historic time for Aston Villa as a club.
"With a guide of £50,000 for this unique collection, we hope football fans and collectors will recognise the rarity and significance of these pieces, and hope they attract significant interest from around the globe.
“Gary’s legacy on the field speaks for itself.
"These pieces aren’t just memorabilia; they’re a tribute to a player who helped define a golden era for Aston Villa.
"We are honoured to offer this unique collection to the public.
"Gary's wishes were ideally for the club to purchase and display for supporters or to go to a Villa fan who will appreciate and cherish them.
"Although the medals would no doubt achieve more being sold separately, the vendors have kept their promise to Gary in keeping the collection as one.
"The collection is offered to the public in a rare opportunity to own a true piece of football history."
Shaw, who was born on 21 January 1961 in Birmingham, made a lasting impact on the sport during his career in the early 1980s.
He was part of the squad during the club's celebrated 81/82 European Cup victory and attracted admirers from fans of other clubs aside from Villa.
He was known for his keen eye for goal, his dedication on the pitch, his sportsmanship and arguably the most decorated player in Aston Villa’s history.
His time at Villa in the early 1980s saw him cement his place as a fan favourite, helping the club during one of its most successful eras.