Djokovic defeated Roberto Bautista Agut 6-4 6-4 to progress to the last eight of the ATP 1000 event in Rome, where he will face Del Potro for a spot in the semi-finals.
The Argentine defeated seventh seed Kei Nishikori, 25 places above him in the rankings, 7-6 (7-4) 6-3 and will prove an awkward obstacle for the Serbian, who continues to struggle for consistency this season.
"I have been doing my best to optimise my game and get it to the highest possible level," said Djokovic, who surprisingly parted company with his coaching staff earlier this month.
"I'm aware that the previous months haven't gone the way that I wished in terms of results, but I believe that the process had to happen."
In other matches, there were straight-sets wins for Marin Cilic, Milos Raonic and Alexander Zverev, accounting for David Goffin, Tomas Berdych and Fabio Fognini respectively, the latter proving unable to build on his upset win over Andy Murray.
John Isner, meanwhile, got the better of 2008 finalist Stan Wawrinka, also in straight sets, while Dominic Thiem came from a set down to get past Sam Querrey.
The biggest casualty of the day was US Open champion Wawrinka, American Isner surprising the world number three with a 7-6 (7-1) 6-4 victory.
The Swiss was broken just once across the two sets but could not make inroads into the powerful serve of his towering opponent, saving match points in vain late in the second before bowing out.
Nadal has been back to his imperious best in this clay-court season, winning in Monte Carlo, Barcelona and Madrid as he eyes yet another French Open crown.
The Spaniard was too strong and too clever for Jack Sock, winning 6-3 6-4 to reach the quarter-finals.
Everything that made Nadal the world's best at his peak appeared to be on display against the outclassed American, as that familiar combination of dynamic court coverage, dogged defence and booming top-spin forehand got the job done with relative ease.
"There is always room to improve," Nadal said.
"That's why I wake up every morning. I go to practice with the motivation to improve something."
As if we needed proof that life is not fair, Thiem's reward for winning the match of the day is a quarter-final against Nadal, the man who defeated him in the final in Barcelona and Madrid.
The Austrian recovered from losing the first set to beat Querrey 3-6 6-3 7-6 (9-7), saving three match points before finally coming out on top in a thrilling tie-break.