Rexach, known as 'Charly', was pretty much woven into the fabric of Barca, playing for the club, taking charge of the first team and playing a prominent role in scouting and it was on one such trip that he would change the life of a young South American.
"I was in Argentina when they first told me about this lad called Messi," he told 'The Sun'.
"But I thought they were talking about a boy who was maybe 18 or 20. I thought, since I'm in Argentina, I'll take a look at him.
"But of course, when they told me that the lad was 12 years old, I was a little surprised."
Nevertheless, the youngster was brought over to Spain and with fears existing over his size owing to his stunted growth, a friendly was organised against much bigger, much stronger players in order to give the youngster a true test.
"I got there at five o'clock and I looked and I walked about a little bit, I looked again," Rexach recalled.
"Football is a team game, right? But this kid could play all on his own, he took the ball and beat other players and scored a goal.
"He had abnormal ability. All instinct. He was born to play football.
"So when I got back to the bench I said 'we have to sign him! That kid is from another galaxy'.
"Physically he was very small - a tiny little thing - but when you saw him play, he was something else. The same as he is now."
Depsite the young Messi impressing Rexach, the player's father Jorge quickly became frustrated by Barca's failure to offer his son a contract, leading to Rexach coming up with a creative solution.
"His dad, logically, got a little angry. He told me that he was going to leave because he felt they were fighting him," 'Charly' explained.
"I gave him my word that we were going to sign his boy and as proof I would sign a piece of paper.
"So I asked the waiter and he brought me ... a napkin!
"So I wrote that, as the technical director of the Football Club Barcelona, I was committed to signing Leo Messi and I signed the napkin."
Almost two decades later, Messi has now won a plethora of titles and team and individual awards, replaying Rexach's faith in him and then some.
"When people hear Messi's story it seems like a fairy tale," Rexach admitted.
"Yet to get there he really had to suffer. But this suffering made him stronger.
"It would be very easy for me to say that I knew what he'd become. But no. What I saw was that Messi was a boy that, if nothing happened to him - if he didn't go crazy or decide not to play football anymore or get badly hurt - he would surely become a player of the highest level.
"But the football played by Barcelona has been perfect for him too. Hunger has come together with the desire to eat. And of course, it is natural that this combination over the years has become explosive.
"If you add Messi with Barcelona and Barcelona with Messi, then this is the result."