Eight-time champion Roger Federer survived a first-set scare to see off South African debutant Lloyd Harris to reach the Wimbledon second round.
Federer, 37, moved through the gears to win 3-6 6-1 6-2 6-2 on Centre Court.
The Swiss, aiming for a record-extending ninth men’s title, will face Britain’s Jay Clarke next.
Third seed Rafael Nadal faced fewer problems against Japanese qualifier Yuichi Sugita, winning 6-3 6-1 6-3.
That set up a tantalising meeting with controversial Australian Kyrgios, who beat compatriot Jordan Thompson in an entertaining five-setter earlier on Tuesday.
Nadal accused Kyrgios of “lacking respect” after the 24-year-old beat him in Acapulco earlier this year, with Kyrgios describing the 18-time Grand Slam champion as “super salty” in a recent podcast.
Asked about the attention surrounding their relationship, Nadal said: “Being honest, I’m too old for all this stuff. What I said, I said. That’s all.”
“I’m not a guy who will be in a fight with anybody. I will keep doing things I love. On Thursday I will be back in this amazing place.”
“At the beginning of this match it was tough. There are lot of things to improve.”