Boulter said she is is expecting “an absolute battle” against her Billie Jean Cup team-mate Dart.
“Playing a Brit in the UK, on the grass, is never an easy draw,” added Boulter.
The pair faced each other at the Nottingham Open last month, with Boulter winning the first-round tie in three sets.
Speaking before Boulter’s victory over Maria, Dart said: “We’re team-mates. We played big tournaments. We’ve grown up together. We’ve been close.
“It’s always going to be tricky, whoever I play, especially when you play a fellow Brit. But it’s great to see that all of us are doing well and putting us in positions where we do get to play each other, hopefully more so in the latter rounds.”
World number 94 Dart did not lose serve against Bai, breaking the 21-year-old in the 10th game to take the first set, and only a rain interruption delayed her progress.
“The last few weeks I’ve had some really good matches, some that have gone my way, some that haven’t,” Dart said.
“For me it’s just being able to build on it, try and play some good tennis. Today I showed that. I was really clinical out there – I was pretty proud of that.”
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