Elsewhere, Fabio Fognini was roared to victory by a partisan Italian crowd as he upset sixth seed Dominic Thiem.
Nadal later eased through 6-1, 6-0 against Bosnian Damir Dzumhur as he targets an eighth Rome title and first since 2013.
"After a loss in Madrid, it's important to come back strong".
"It was a positive match, of course", said Nadal.
Djokovic was up against Georgia's Nikoloz Basilashvili and settled down after a nervy start to wrap up a 6-4, 6-2 victory after one hour and 18 minutes.
However, before they can look ahead to a blockbuster encounter, Nadal will first have to get past Dutchman Robin Haase/Canadian Denis Shapovalov, while Djokovic next faces Spain's Albert Ramos-Vinolas, who defeated American John Isner yesterday.
Fognini, ranked 21, next plays Germany's Peter Gojowczyk before a possible quarter-final showdown with top seed Nadal. So I'm disappointed that I lost, obviously.
"It was a good match, actually, from both of us, I would say", said Thiem.
With the score at 6-3, 3-6, 5-5 and serving at 30-30, Pliskova had an overhead smash incorrectly called out, and after the umpire, Marta Mrozinska, could not find the ball mark the original call stood. She could not hide her fury as she went on to lose the match 3-6, 6-3, 7-5.
Instead of shaking hands with the umpire after the match, Pliskova first put her hand out before beating up the chair.
In the women's event, world No 1 Simona Halep was in ruthless form against Japan's Naomi Osaka, the Romanian needing just 59 minutes to win 6-1, 6-0 to reach the third round. "That was the way that I want to play".
The former World No 1 needed to save four break points during a marathon seventh game but clinched a break of his own a couple of games later to go one set up. And that's what I did today.
"I don't know how many mistakes I made, but not many".
Girl, 11, dies after tree falls on vehicle in Newburgh
The Hartford Courant reports more than 90,000 remained without power Wednesday morning and "scores of roads are impassable". In Danbury, Conn., a teenager suffered serious injuries after being hit by the roof of a baseball dugout.
Kickoff set for Alabama opener
Alabama's 2018 opener against Louisville in the Camping World Kickoff Classic in Orlando, Florida, will kick off at 7 p.m. The last pairing came on January 1, 1991 with the Cardinals claiming a 34-7 win in the Fiesta Bowl in Tempe, Ariz.
Sherpa climbers set new summit records on Everest
Summits are underway every day since May 13 after the eight rope-fixing team members opened the climbing route for all climbers. A Sherpa guide has been missing on Everest's southern side in Nepal since Monday, said Mingma of Seven Summits.