Brundle was coming to the end of his usual pre-race grid walk when he reached a roped-off section at the front of the starting grid which was only open to selected VIPs.
Among them was tennis legend Roger Federer, who had been one of Brundle’s targets from a list of attending VIPs.
Brundle happened to come across Stewart, who with his Swiss connections knows Federer well. The three-time world champion agreed to duck under the rope and ask 20-time Grand Slam winner to come over and talk to Brundle.
However, security personnel decided that Stewart shouldn’t be there and appeared to be ready to throw him out before he was able to get Federer's attention and escort him across to Brundle.
The incident has gone viral on social media, with some observers suggesting that the treatment of one of F1’s living legends reflects the championship’s move from its roots to cultivating showbiz connections.
“What a legend, 83 years old,” said Brundle on the Sky Sports F1 podcast. “I happened [to come] across him at the front of that rope.
“It was like trying to get into a dodgy nightclub, wasn't it? The pair of us, we had no chance, we had the wrong dress code or something.
“I find that whole procedure a little bit weird on the grid. There were so many people around, and I particularly wanted to talk to Roger as I got to the front there.
“I just happened to say that to Jackie, and he was like, 'do you want to talk to him then?' And I said ‘Yeah.’ ‘Well, I'll get him, I'll get him.’
“He jumps under the rope, and now there's three people running after him. And I'm like, ‘Oh my God, I've just got Jackie Stewart thrown out of this place.’ We have both been thrown out of better places than that! It was hilarious, really.
“Then I started to panic a little bit, if I'm honest. He's 83, I don't need him over there risking himself just for an interview. He was great. That's a true mate, isn't it, when they go the extra mile for you like that.”
Brundle said there was even more stress on his side as he was counting down to the end of the time allocated for his grid walk before coverage switched to the presentation of the drivers.
“I was in, what we call, a hard count in our business into throwing to the presentation,” he explained. “It's sort of non-negotiable. Now I'm watching security guards, Roger Federer, then George Russell gets involved trying to explain what's going on. And it's Sir Jackie Stewart!
“I see this negotiation going on over there. Literally, my count in my ear just finishes as Federer turns around and walks over, escorted by Sir Jackie, so I think I've got to talk to him now. I think we got away with it somehow.
“But huge amounts of anxiety, that's why I never watched the damn things. And I haven't watched it for 26 years.
“I've seen it on social media, that bit, but I never watch the grid walk back ever. I don't know who that bloke is running around being cheeky and rude, my sort of alter ego!”