The American, seeded 29th, won in straight sets against France’s Kristina Mladenovic before watching the British number two topple eighth seed Petra Kvitova in three sets.
“We’re the same age, I guess it’s a rivalry. I mean it’s not like Federer-Nadal rivalry,” Stephens said. “It could be. We’ll see.”
Robson lost the first set but battled back against the 2011 Wimbledon champion to win 2-6 6-3 11-9 in just under three hours.
The 18-year-old suffered a 6-4 7-6 (4) loss to Stephens at Hobart earlier this month, but Robson is confident she can improve on that result.
“We first played each other when I was 10 or 11,” said Robson.
“She beat me last week in Hobart, and it was a pretty close match, but I definitely feel like I could play a lot better than I did last week, and it’s going to be really tough.
“She’s in good form at the moment. She’s a good mover and a good ball striker. It’s always going to be a tough match but hopefully I can play better than I did today.
Robson also believes Stephens could be a Grand Slam winner in the future.
“I don’t see why not. It’s very easy to see how naturally well she hits the ball and how easy everything sort of comes to her.
“She’s always been good, so it’s not a surprise that she’s doing well.”
MORE: The latest football news
MORE: The latest tennis news
BIOGRAPHY: Unai Emery
BIOGRAPHY: Adam Lallana