Williams beat Graeme Dott 10-4 to reach the last 16 and is the last Welshman left at the Crucible.
Holt beat former world champion Neil Robertson 10-6 to meet Williams and the Welshman said: "There probably won't be too many handshakes with me and him.
"It'll probably be more hugs and kisses before we start!"
Williams is in the last 16 for the first time since 2012 while Holt is there for the second time.
"If I can keep playing the way I'm playing I'm going to be a tough man to beat," said Williams, who played down his chances ahead of the tournament.
"I'm not saying I'm going to win the tournament."
Williams was not surprised qualifier Holt beat 2010 champion Robertson in the first round.
Robertson was the third of the world's top four to lose in the opening round following defeats for defending champion Stuart Bingham and Shaun Murphy,
Former double world champion Williams says those results are a clear indication of the strength of competition at the highest level of the sport.
"Top players have gone out for a reason," Williams told BBC Sport.
"It's because the other players have played better than them. There are no shocks really.
"People might think that Michael beating Neil was a shock, but if you're a snooker player it's not a shock.
"Everyone is capable of beating everyone else."
Williams will be aiming to reach the quarter-final at the Crucible for the first time since 2011.
He is relishing the prospect of taking on a player he knows well after Holt's impressive 10-6 victory over Robertson.
"I didn't see much of the match, but I saw how many centuries and big breaks Michael got," Williams added.
"He obviously played well and deserved to win. I'm looking forward to it because he's a good friend of mine."
Mark Williams believes he is "a tough man to beat" at the World Championships as the double world champion faces "good friend" Michael Holt on Friday.