Bielsa has secured legendary status at Leeds United having led the Whites back to the Premier League after a 16-year absence.
The Argentine appears happy in Yorkshire and it would be a surprise if he were to depart before their top-flight return.
But Bielsa is out of contract at Elland Road this summer and has yet to sign a new contract, raising fears he could be tempted away.
And, according to The Sun, Messi wants Barcelona bring his fellow countryman to the Camp Nou.
The Spanish giants are in turmoil having relinquished their LaLiga title to Real Madrid and with boardroom politics and financial issues having a damaging effect on the club behind the scenes.
Quique Setien was only appointed manager in January but is understood to have a strained relationship with senior players, including Messi, and could now be set for an early exit.
Messi carries huge influence at Barcelona and recently derided his teammates after their title failure.
"We are a weak team who can be beaten with enough intensity and enthusiasm," he said. "We have lost a lot of points which we shouldn't have done [and] been very inconsistent.
"We need to be self-critical, starting with the players but also across the entire club. We are Barcelona, and we are obliged to win every game."
Messi is likely to have a huge say on who will take over should Sieten depart and is said to feel Bielsa would perfectly fit Barcelona's ideals.
Pep Guardiola, with whom Messi enjoyed so much success at Barcelona, has spoken of his admiration for the 65-year-old's coaching methods.
"He is unique in world football because of the special way he plays," said the Manchester City manager recently.
"I learnt a lot about his style, his final product. He's an incredible person, so special."
Bielsa has managed in LaLiga previously, with Espanyol and Athletic Bilbao and has huge experience, having also managed the national teams of Argentina and Chile.
The Sun say Barcelona have to make contact with representatives of Bielsa but the threat of a move 'will set alarm bells ringing at Elland Road'.
Lionel Messi wants Marcelo Bielsa to become Barcelona's new manager.