The Serbian, who has struggled with injury for much of the past year, was asked when he planned to make his first grass-court appearance.
No. 13 Madison Keys also plays on Tuesday against Yulia Putintseva.
Her next outing will be a battle of wills against compatriot and friend Stephens, who has emerged victorious in both of their previous meetings as professionals, most recently beating Keys in the final of last year's U.S. Open.
Djokovic, seeded 20 after a year of struggles with an elbow injury and mediocre form, was never in trouble against an erratic Verdasco, who needed treatment for a badly blistered left foot at the end of the second set.
"I think that's going to be the number one priority for me now, to kind of step out from the tennis for a little bit, try to watch some matches and sort of try to progress somehow and just get better".
"He is one of the fittest guys on the tour so it was hard for him today", said Thiem, the only man to have defeated 10-time French Open champion Rafael Nadal on clay this year.
He was suspended for 18 months by the Italian tennis federation in July 2016 before later being cleared of any wrongdoing.
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After fighting through three consecutive five-set match-wins, Zverev is still on track to be the first German men's champion at Roland Garros since 1937.
But the Russian sealed the set at the third time of asking, leaving the world number three needing a fightback for the third consecutive match.
Will Halep make it third time lucky?
"Once you get to the second week, every match obviously has more nerves, and there's more on the line and all of that", said last year's US Open runner-up.
Yet Djokovic won the third set at a canter, and a suddenly frustrated Cecchinato was hit with a point penalty for unsportsmanlike conduct.
But the 10th seed will have to be wary of the risky Kasatkina, who saw off Caroline Wozniacki in straight sets to notch her third consecutive win over the Australian Open victor.
"When you want to be the best player in the world and have been the best player in the world, you hate the defeats".