Conceicao looked to get angry with the German after the pair fist-bumped at full-time, with Chelsea having secured their place in the semi-finals, despite losing 1-0 in the second leg.
The Portuguese manager was seen speaking to the fourth official, as well as his own players, who then also began to argue with Tuchel.
Porto defender Pepe then appeared to square up to the Blues manager, with one of the officials having to stand in the way of the pair.
Players of both teams also had to be separated as they came together on the pitch at the final whistle.
Conceicao refused to specify what was said, but suggested his opposite number was far from polite. After the game, Conceicao claimed: “I was insulted by this man who is next door.
“My English is not perfect, there is no point in talking about it. I was focused on the game; I don’t know why he reacted.
“I heard some insults, but it’s gone … it’s not pretty and my irritation in the end had to do with it.”
Tuchel downplayed the incident, and the Blues went on to deny their manager made any insulting remarks.
“It was just a little argument with me and their coach, and everybody was following me suddenly,” Tuchel said. “No harm done.”
Both bosses had also clashed on the sideline during the game when Porto were denied a penalty.
Chelsea will now go on to meet either Liverpool or Real Madrid in the semi-finals.