Zampa, who will be leading Australia's spin attack in the tournament with Glenn Maxwell as his sidekick, wore a cut on his face during training after a mishap in the swimming pool.
"He swam into the pool wall apparently," Cummins told reporters on Saturday.
"He said he had his eyes closed and thought he was swimming in a straight line and swam into the step in the pool... He's all good he's just a little bit sore."
It must have appeared familiar to Cummins, who filmed an absent-minded Alex Carey, their preferred stumper in the World Cup squad, walking straight into a hotel swimming pool in Karachi last year.
All-rounder Maxwell himself missed the entire 2022-23 summer after a freak incident in which he got his leg broken at a birthday party where he was fooling around.
More recently he impressed with his bowling in the ODI series against India and will be confident of troubling batters in conditions that suit his craft.
"We've seen Max, he's a front-line spin bowler," Cummins said of the all-rounder.
"In the 2015 World Cup, he was the sole spinner in basically every single match I think other than one, so really happy with how he's going.
"I thought he bowled really well in that third ODI against India (last month). So yeah, we've got 20 overs of spin out there if we need it."
Australia's batters will have their task cut out in Chennai, where conditions usually assist spinners, and left-handed opener David Warner may have provided a glimpse of that when he tried a radical approach in the second ODI last month.
In that match in Indore, Warner briefly flipped his batting stance to right-handed to try and negate off-spinner Ravichandran Ashwin.
Cummins said his batters would have their plans against the Indian bowlers - though it may not involve more ambidextrous batting.
"They're a good bowling lineup, especially here in home conditions. So they're going to be challenging," Cummins said.
"The good thing is we've played them a lot. So our batters will have their own plans."
"Davey, we'll see if he bats right-handed or left-handed. He's probably better left-handed, I reckon, but we'll see how we go."