The Scot, who managed the Reds for three seasons after six years as a player, received a letter while in charge of the first team from a young Schmeichel, but opted against pursing the goalkeeper having recently signed David James.
Schmeichel went on to join United shortly afterwards for around £500,000 and won five Premier League titles and the Champions League once, among other honours.
"Schmeichel wrote to me when I was at Liverpool in the early 1990s," Souness told the Daily Mail. "Ron Yeats came into my office and told me there was a young Danish goalkeeper who was a Liverpool fan and was willing to pay his own travel and hotel in exchange for some time with us.
"But at the time I was trying to ease Brucey Grobbelaar out and that was proving a hassle. And I think I had just signed David James. So I thought I could do without it."
Another star Liverpool missed out on was Cantona, with Souness rejecting advice from Michel Platini to sign the "problem boy" who went on to lead rivals United for five years, winning four titles.
"Similar with Eric," Souness continued. "We had played Auxerre at home and Michel Platini came to see me. He said he had a player — a problem boy but a proper player. Cantona. I said the last thing I needed was another problem player."