He went on to manage Marseille and Lazio, and his last job was manager of Lille in 2017, where he only lasted seven months after a run of just three wins in 13 games.
Bielsa, 62, would be a stunning replacement at Elland Road and a well-placed source has told Football Insider that Leeds have held face-to face talks, and that owner Andrea Radrizzani is personally spearheading the move.
That may have been harsh on the 40-year-old, who never even had a transfer window with which to try and mould his squad, but after a disappointing end to the campaign he has become the latest in a long line of sacked managers in recent times.
Heckingbottom's assistant, Jamie Clapham, head of fitness Nathan Winder and analyst Alex Bailey will also depart, while the set-piece coach Gianni Vio will not have his contract renewed.
Heckingbottom will no doubt be disappointed at the decision - considering that he was only given the back-end of the season - however, this is trigger-happy Leeds we are talking about.
The club's managing director Angus Kinnear admitted he wanted to bring in a "head coach with more experience".
"Paul came to us during a hard period in the season and has conducted himself in an exemplary manner despite results not going as any of us had hoped".
Having managed Velez Sarsfield and Newell's Old Boys in his homeland, he coached his national side for six years as well as the Chile national team. "We are confident of making a quick appointment and we thank our fans for their continued support".
He quit Barnsley, his boyhood club, to take over at Elland Road after exactly two years in charge at Oakwell.