The playmaker held talks with the Anfield hierarchy in 1983, with Shoot magazine even reporting that the deal had been done.
Speaking to Liverpoolfc.com last year, Laudrup explained: "I had the opportunity to sign for Liverpool. We actually agreed a deal and I came over with my father to talk to Liverpool's representatives.
"We agreed a three-year contract and went away thinking it was all done. 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."