Arsenal have also been in the running for his signature – and a move to London does make sense.
He would slot in beautifully to Mikel Arteta’s midfield – giving him the versatile playmaker he’s been searching for.
But it looks like Old Trafford is a more attractive destination for him – meaning the Gunners look likely to be gazumped.
But what exactly are they missing out on?
Tolisso developed in Lyon’s academy, spending four years in the first team there – playing alongside former United star Rafael da Silva, who was quick to tweet his enthusiasm for him.
“One of the best midfielders I’ve played with. I think he will be perfect for Manchester United. Hope he goes there,” Rafael declared on social media.
In his last season in France, Tolisso scored 13 goals in all competitions, often as an attacking midfielder slotting in behind his contemporary Alexandre Lacazette – with France international Jean-Pierre Papin likening him to Toni Kroos in his style of play.
“He can defend, he can attack and he can play an important role in midfield. He has great technique, is dangerous in front of goal and can also make goals,” Papin said to Bild in 2017.
Tolisso himself acknowledges most of his impact comes from his ability to pass – he is not the kind of player to take defenders on.
He thinks the comparison to Kroos is fair – and also says he admires players such as Luka Modric, Thiago, Steven Gerrard and Frank Lampard.
“I try to be as effective as possible,” he has said. “Control, pass, control, pass, move, then to be decisive – assist, goal.”
He says that he is working on becoming a more dangerous prospect in front of goal.
“I love scoring and setting up goals,” he told Sky Germany. “That’s very important for my playing style and something I’m measured on.”
Tolisso cost Bayern a club record fee of £37million three years ago, and enjoyed a very successful 2017-18 campaign as he was almost ever-present in the side – but has a troubled time since then.
The 25-year-old won the World Cup with France two years ago – and then suffered a cruciate ligament injury in his right knee, keeping him out for the majority of the 2018-19 season.
He has only made 15 starts so far this season, scoring three times – but he has also had to have an operation on his ankle ligaments, meaning there could be question-marks over his fitness.
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And he’s struggled to make himself a first choice following his injury woes.
“At a club like Bayern, you have to be a great fighter,” reflected Bayern legend Bixente Lizarazu earlier in the year. “Tolisso is a very good player. If he believes he doesn’t have a chance and doesn’t feel good, it would perhaps be better to try out something new. That sometimes happens in a career.”
And if Tolisso does decide his future lies elsewhere, Bayern might be prepared to let him go – especially having signed Leroy Sane from Manchester City