In a repeat of their epic Australian Open final, the world number one will face the Spaniard in the Mexican Open semi-finals.
Norrie defeated Gojowczyk 6-1 6-0 to reach the last four.
The new world number one, Medvedev, defeated Nishioka by a score of 6-2 6-3.
Nadal defeated Tommy Paul in the quarter-finals.
The 35-year-old Spaniard won a record 21st men's Grand Slam title by beating Medvedev in a gripping Melbourne final last month.
The Serbian will become the world number one for the first time after he lost to the Czech.
It is not easy to play a match when you get this news during the day, as was the case with Medvedev, who found out he would take over from Djokovic.
The first goal for me was to win today, because I want to win every match I play. It is definitely some great news.
The more serious news of a full-scale invasion on Ukraine has made it hard for Medvedev to celebrate.
It was not easy to wake up in Mexico after watching the news from home.
I want to promote peace by being a tennis player. I have been in many countries as a junior and as a pro.
It is hard to hear all this news. I am all for peace.
After dropping just 10 points in a dominant opening set against Paul, Nadal went a break down early in the second but there were five more breaks of serve on the way to the tie-break.
Paul led 2-0 before Nadal came back to set up a meeting with Medvedev on Friday.
Nadal said that he played one of his best sets for a long time in the opening set against Paul.
Everyone knows how difficult it is to play against Daniil. If I want to have a chance, I have to play at my highest level.
Norrie was in fine form by breezing into the last four against Gojowczyk.
Gojowczyk reached the last eight with a walkover when Alexander Zverev was kicked out of the tournament for hitting an umpire's chair with his racquet.
Norrie, ranked 12th in the world, had a lot of quality as he looks to win back-to-back titles following his triumph in Delray Beach last week.
The Greek world number four will play Norrie in the semi-finals after he defeated American Marcos Giron.
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