This was Billy Bremner, aged 13 in 1956, with the 15-year-old Smith attached at the time to his local u-21 team, Gowanhill United.
"We had just finished training and were about to have a game," Smith recalled. "Then this wee fella, who I didn't know but seemed to know me, shouted over: 'Alex, any chance of joining in?'.
"I said I'd ask our trainer, George McDonald, but George looked at Billy and said: 'He's too wee, we don't want to take any chances'.
"The boy was desperate, though, and we only had 15 players. So I asked again and George finally said: 'Well, okay'.
"Billy ended up on the same side as me and was unbelievable, the best player on the park. At the end, he said: 'See ya again' and, eventually, he signed for us".
So began a friendship which was to endure until, December 7, 1997, when Bremner died of a heart attack just two days short of his 55th birthday.
Smith can remember not only the day, but the hour and where he was, when he received the news.
"It was 12.30pm on a Sunday, with all the SFA coaches gathered at a seminar organised by Frank Coulston at Jordanhill College," the former Aberdeen boss said.
"I was sitting with Asa Hartford when Frank came over with the message, saying there was a posse of pressmen outside wanting to speak to me. Billy had a couple of flutters before taking a heart attack on the Friday night. He then seemed OK, only to be hit again on the Sunday.
"I was really shocked. Our friendship had been cemented from that one night those years earlier. Billy was a great companion and a great pal."
And a great player, too, winning 55 caps with Scotland, whom he captained at the 1974 World Cup Finals, as well as skippering Leeds to a host of honours.
"I scored 72 goals one season with Gowanhill and Billy must have made the half of them," Smith attested. "I also remember seeing him in a schoolboy trial. It was blowing a blizzard but what a game he had. He went on to play for the Scottish Schools when they won the Victory Shield at Wembley.
"Billy was the best I saw at that level. He could do the lot and went straight from school to Leeds at 15. He made their first team when he was 16 against Stoke, who had Stanley Matthews playing for them.
"Leeds went down that year (1960), then Don Revie came in and got them back up again."
Revie is to be credited, too, with talking a teenage Bremner out of leaving Elland Road.
"Billy was unsettled and desperate to come up the road again," recalled Smith.
"Celtic were interested, as were a lot of others. But Revie convinced him he should stay. Then Billy met a Stirling girl, Veronica Dick.
"They got married when he was 19. It was the best thing that could have happened because it settled him down and let him get on with playing football."
Smith, best man at their wedding, went on: "I've never known anyone to have such an influence over a group of players as Billy had at Leeds. Jackie Charlton was six or seven years older but Billy was the captain. He had an unbelievable desire to do well.
"I remember him saying: 'Alex, I never go on the park thinking anyone is better than me'. He played for Scotland against Brazil (in the Maracana) in 1972 and finished up man of the match.
"Outside Wembley once, before Leeds were playing, I was waiting for Billy to give me tickets. He didn't appear until 15 minutes before kick- off and, when I said: 'You're leaving it a bit late', he said: 'Ach, it's just another game'.
"Billy was a fantastic captain for Scotland, as well. It was a tragedy they didn't qualify for the later stages of the 1974 World Cup."
Highly unfortunate, too, that Bremner's international career came to an inglorious end the next year when a misadventure in Denmark resulted in the so-called Copenhagen Five, Bremner among them, being banned by the SFA.
"It was just one of these things," said Smith. "The players had a night out after the match, as had the officials. Billy could have gone on to reach 70-odd caps. But, by the time the ban was lifted, it was too late for him to come back."
Smith, having revealed that one of Bremner's unfulfilled ambitions was to manage Scotland, went on: "I remember seeing Billy play for Leeds in a European tie against Hibs at Easter Road.
"He was at sweeper and, at one point, actually kept the ball up on his own goal-line. Unbelievable, it was. What a player! In my eyes, he was the best ever, but I'm biased."
Biased in favour of the boy from Raploch who never forgot the favour Smith did him by getting him a game with Gowanhill.