Katsuki and Tamaki do not know each other and have not been seen together or speaking in Manga canon because when the Big Three appeared before the Class 1-A students, Katsuki was still being punished for his fight against Izuku.
However Tamaki seems to know who Katsuki is thanks to Kirishima. In School Briefs vol.4 Kirishima and Tamaki were patrolling and they mention the fight between Izuku and Katsuki Bakugo the other day and then Kirishima starts sharing Katsuki characteristics with Tamaki and says that Katsuki is just the type of guy to tell you how he really feels and be honest with himself.
During the filler Episode of Season 5, the students of Class 1-A are in a training exercise in which Tamaki and Nejire acted as villains against Class 1-A while Mirio acted as Civilian in distress. After Tamaki defeats Izuku, Kirishima, Katsuki and Denki appear and Tamaki and Katsuki briefly face each other (being their first interaction in the Anime) until Tamaki is scared of Katsuki's angry face and Katsuki takes advantage of this to derive it.
Despite the victory of Class 1-A in the drill and the congratulations of his teammates for taking down Tamaki, Katsuki still about Tamaki, says that he hasn't finished yet and Tamaki is surprised as Katsuki angrily tells him that he held back on him. Before Tamaki could answer him that his angry face scared him, Katsuki yells not to underestimate him and even if he was acting like a Villain, he is a Villain and is his enemy and then put his hand on Tamaki's face and throw a big explosion, while Tamaki with tears in his eyes says internally that he should have gone home.
TamaBaku is amongst the rarest of pairs due to their lack of interactions, and the absolute difference between them. However, a few fans have found their potential dynamic interesting enough to work, and commonly portray them as meeting through Kirishima.
A popular headcanon among TamaBaku and KiriBaku fans is for Bakugo to be jealous of Tamaki's closeness to Kirishima and ignore or confront and fight Tamaki about it. Sometimes the headcanons usually end with Bakugo realizing that Kirishima was never in love with Tamaki or fixing things with Tamaki and befriending him and finding out that there was no romance between Tamaki and Kirishima.
Some fans speculate that in the future, Bakugo and Tamaki will work together as Izuku, Bakugo, and Shoto are considered to be U.A.'s next The Big Three and Izuku and Shoto have already worked with Mirio and Nejire at some point. The theories have been made on the basis that some fans have realized that Izuku and Shoto have already interacted and fought side by side together with a member of The Big Three in long and important arcs (Izuku together with Mirio during the Shie Hassaikai Arc and Shto alongside Nejire in Paranormal Liberation War Arc) and suspect that Bakugo and Tamaki will interact and fight side by side in a possible major arc of the manga.