Earlier, Halep powered her way to a 6-1, 6-0 win in just under an hour on the clay of the Italico Foro to claim her fourth win over 21st-ranked Osaka in five meetings.
Nadal, who lost his number world one ranking after his 14-match winning run on clay was snapped by Thiem in the Madrid Open quarter-finals last week, was back to his best as he wrapped up victory with a blistering forehand victor.
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.
"It was important, after a loss in Madrid, to come back strong, and that's what I did today", said Nadal following his demolition of Dzumhur.
"I beat a top clay-courter", Fognini added.
"I just really wanted to win this match against a really classy player", said the five-time Grand Slam champion. And I knew she knew I couldn't hit a backhand, so we were just having backhand rallies.
Fognini had his home-country crowd behind him, and when he went three set-points up against Thiem, it was clearly too much for the Austrian to take. "It was one of those helpless feelings", conceded 20-year-old Osaka who could be heard saying "I want to cry" courtside.
Nadal will next face Canada's Denis Shapovalov, who faced a grueling test in battling past Dutchman Robin Haase 7-6 (3), 6-7 (5), 6-3 in almost two and a half hours.
EPA-EFE/ETTORE FERRARISerbia's Novak Djokovic in action during his men's singles second round match against Georgia's Nikoloz Basilashvili at the Italian Open tennis tournament in Rome, Italy, 16 May 2018.
Nadal is aiming for an eighth title in the Italian capital this week, while Djokovic will hoping for a solid performance after suffering six defeats in his 12 tour matches this year.
Also on the red clay, Kei Nishikori rallied to beat third-seeded Grigor Dimitrov 6-7 (4), 7-5, 6-4.
Defending champion Alexander Zverev beat Italian wild-card entry Matteo Berrettini 7-5, 6-2. Fifth seed Juan Martin del Potro emerged a 7-5, 6-3 victor over Barcelona runner-up Stefanos Tsitsipas and Miami Open champion John Isner was edged 7-6 (5), 6-7 (2), 6-7 (5) by Albert Ramos-Vinolas.
Thiem, who finished runner-up in Madrid on Sunday, struggled to find his rhythm in a 6-4 1-6 6-3 defeat by Fabio Fognini.
Nishikori dropped the first set but mounted an impressive comeback to secure a hard-fought 6-7, (4-7), 7-5, 6-4 victory.
He will now take on British number one Kyle Edmund, who saved the only break point he faced in a 6-2 7-6 (3) victory over France's Lucas Pouille.