In 1983, the Brondby star wanted to join them, he was a Liverpool fan, but ultimately the transfer didn't materialise over a disagreement and England's loss was Serie A's gain.
"We agreed a three-year contract and went away thinking it was all done," Laudrup explained in 2012. "But two weeks later they came back saying they wanted to offer me the same package, but for four years because I was still young and needed time to develop.
"I was disappointed because although nothing had been signed, we had an agreement. I decided not to join them and people thought I was crazy. This 19-year-old was turning down the great Liverpool. It wasn't really like that though.
"I just felt an agreement is an agreement and people should stick to what they decide. After that, I had other offers and went on to join Juventus. That's football for you."
Laudrup joined Lazio from Brondby after which he played for Juventus - a rival of Liverpool's in Europe at the time - Barcelona and Real Madrid. This is a man three-time Ballon d'Or winner Michel Platini called one of the most talented players he has ever seen.
READ MORE TRANSFERS
Hijacked
Arsenal pulled off greatest hijack when they got Spurs to pay for player's taxi
Spurs hijacked Man United move for top-rated midfielder by buying his family a house
His brother, Brian, once said: "He started as an attacker but became an elegant attacking midfielder, perhaps the most complete there has ever been.
"His vision, speed of thought and passing were on a different level; he always knew what was going to happen before anybody else did. If anyone had a 'football brain', it was him."
Romario, Raul, Luis Figo - all of whom have raved about the player Laudrup was.
Figo puts him among his favourite players. "When I was younger I also used to enjoy watching players like Michael Laudrup, Zico and Marco Van Basten," adding the Dane was the best he ever faced.
Most read in Football
Brighton still hopeful of signing Chelsea starlet, who fans joke has 'generational aura'
Man United announce Mount, Mbappe given ultimatum, big Rice and Timber to Arsenal updates
Chelsea display at Stamford Bridge looks bizarre as Blues' summer overhaul continues
England U21s one win away from first title since 1984 as Young Lions ease past Israel
In Spain, first with Barca, he won four successive La Liga titles and added another league crown at Real Madrid.
Ivan Zamorano, his Real Madrid teammate, called him a 'genius'.
Johan Cruyff, his manager at Barcelona, loved him. "When Michael plays it is like a dream, a magic illusion and no one in the world comes anywhere near his level."
It's rare for a player to be loved in equal measure by both Barca and Real supporters, but Laudrup is.
One story goes that saw King Juan Carlos I appear at his shoulder when he was pondering his future over some food in a posh Madrid restaurant in 1996.
He had been sitting a few tables away and overheard the conversation, prompting him to walk over.
"That's good news," he whispered in the player's ear when it looked like he was leaving. "I'll be the only King in Madrid again."
Liverpool, though, didn't exactly regret the decision given they won the treble in the 1983/84 season, clinching a fourth European Cup in seven years, but a front line of Laudrup, Kenny Dalglish and Ian Rush would have been frightening.
"Michael turned out to be a wonderful player, but I firmly believe we made the right decision at the time because he was demanding terms and conditions in his contract that were impossible for us to meet," former Anfield executive Paul Robinson said.
"The main stumbling block was that he wanted the option to be able to just walk away after a short period if he felt things weren't working out. From our point of view, this just did not make any business sense.
Read More on talkSPORT
WWE legend Hulk Hogan shows off jacked physique alongside new yoga instructor girlfriend
'It's the first question' - Maddison will have signed for Tottenham on one key condition
"Over the years it was very rare for a player to turn Liverpool down but it wasn't something we dwelt on. It was just a case of moving on and identifying another target."
Oh what could have been.