The defending champion struggled with his form and fitness in a laboured 6-3 5-7 6-2 6-1 win over Poland's Jerzy Janowicz on Monday, adding a right elbow issue to the left wrist injury that has dogged him in recent months.
The left-handed Vesely beat Djokovic on clay in Monte Carlo earlier this year but the Serbian world number one said that defeat would have no bearing on Wednesday's encounter.
"Different surface, different circumstances, best-of-five," said Djokovic, a 12-times grand slam champion who lost early at Wimbledon this year and also at the Rio Olympics this month.
"But still, Vesely deserves respect. He's somebody that has been kind of trying to break through as the next generation.
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"A couple of years ago, he already was there. He made a name of himself. Just gained the consistency, I think, over the last couple of years. He has a big game, a big serve, big forehand, and moves well for his size. So let's see.
"Obviously he hasn't played many times on the Arthur (Ashe) Stadium. Conditions are quite suitable to my style of the game. Hopefully I'll be able to slow his serve down a little bit and then take it from there."
Fourth seed Rafa Nadal, who is recovering from a wrist injury that forced him to pull out of the French Open and then to miss Wimbledon, plays in Wednesday's night session against Andreas Seppi of Italy.
The Spaniard says his wrist is improving day by day but concedes that he needs to cut loose on his forehand.
"When you have pain, you try to change the movement to avoid a little bit that pain," Nadal said. "So I need to find again the normal movement. But I am on the way."
Australian Open champion and second seed Angelique Kerber of Germany, third-seeded Spaniard Garbine Muguruza and Czech Petra Kvitova, a twice former Wimbledon champion, are also in second-round action.
(Editing by Mark Lamport-Stokes)
