The 19th seed is a set up on world No. 11 Casper Ruud on the John Cain Arena.
There have already been plenty of upsets this week, with last year’s finalist Elena Rybakina and No. 8 seed Holger Rune among the victims.
World No. 14 Tommy Paul – a semi-finalist here last year – lost to Miomir Kecmanovic in five sets earlier on Saturday.
And world No. 75 Anna Kalinskaya upset former US Open Sloane Stephens on day seven.
Speaking after she came back to win 6-7 6-1 6-4, the 25-year-old said: “I saw a few upsets, and it just, for me, means that anyone can win. The most important is to fight, and you have to enjoy, and then in the end of the match, you will see how it goes.”
And Norrie could join them if he takes out the three-time Grand Slam finalist.