The Red Devils were reduced to 10 men after midfielder Ander Herrera received two yellow cards in the first half. Following the Spaniard's sending off, the two managers were seen arguing on the sidelines.
Conte replaced Mourinho at the west London club after the latter was sacked from his post in December 2015, just months after guiding Chelsea to the league title. The Portuguese tactician said that his player needed "immediate" attention that saw him refuse to shake hands with the former Juventus and Italy coach, but admitted that he would have shook hands with Conte if he had spotted him on the pitch.
"If I see him here I will shake it," Mourinho explained, as quoted on United's official website.
"I didn't because I thought my players deserved my action immediately and I our supporters also deserved our thanks to them, so it was that. I can promise you that if I see him out there I will shake his hand. I have nothing to speak with him at all, but to shake his hand, for sure I will do that."
Meanwhile, N'Golo Kante's first goal for Chelsea came in their 4-0 league win over United in October. The former Leicester City star was on target against the same opponents and the midfielder's 51st minute goal was the difference between the two sides during the quarter-final clash.
The holders have been knocked out of the FA Cup and the Stamford Bridge outfit have booked a place at Wembley after their latest win. Chelsea take on cross town rivals Tottenham Hotspur in the semi-final clash, while Manchester City and Arsenal face each other in the other semi-final fixture.