Chelsea progressed to the last four of the competition after a 2-1 result on aggregate despite losing 1-0 to Porto on the night in Sevilla.
After the final whistle, Tuchel hugged his coaching staff before walking in the direction of the fourth official and Conceicao.
Tuchel extended his right arm for an apparent fist bump with Conceicao, which was reluctantly accepted, before walking away as the Porto boss exchanged words with him.
"I was insulted by this man [in reference to Tuchel, who was also in the mixed zone giving an interview] that is here beside me," Conceicao told Portuguese TV channel TVI24. "I told the referee that the fourth official heard those insults. That's all."
Conceicao did not reveal the exact words that Tuchel said to him but told his postmatch news conference: "It's no use talking about it. My English is not perfect. There was no exchange of words, I was worried and focused on the game. I didn't understand the reason for the reaction. I heard some insults, it's not pretty, but that is over.
"My irritation at the end had to do with that because I didn't even speak to him during the game, I didn't look at their bench, but at the game. I was worried about the game."
Tension between the coaches were apparent during the early stages of the match. Tuchel protested vigorously to the fourth official and at the Porto bench after an elbow from Porto captain Pepe on Kai Havertz in the third minute, while in the 29th minute, Conceicao complained after Cesar Azpilicueta's foul on Otavio.
Chelsea and Porto players had to be separated after the final whistle, while Pepe was seen angrily pointing a finger at Tuchel. After the match, the Chelsea boss played down the incident.
"It was just a little argument between me and the coach [Conceicao]," Tuchel said at his news conference. "And then everybody was following me suddenly; I thought it was between me and the coach, but obviously it was not. No harm done."