The reigning Masters and US Open champion will supplant injured Northern Irishman McIlroy at the top of the rankings this weekend if he wins the $9.25m event at Firestone Country Club.
However, the 22-year-old's push for top spot faltered with a patchy level-par 70 that saw him finish the day well down the leaderboard, headed by New Zealand's Danny Lee.
Spieth, who has not played since the British Open at St. Andrews last month, blamed ring-rustiness after a round containing two birdies and two bogeys, but is confident he can challenge the leaders.
"The challenge is just getting into it," Spieth said. "I just felt like I started the round today with little to no emotion.
"It's a battle, but it's one that I'm looking forward to because I'm not out of it," he said.
Spieth is using the tournament as preparation for next week's PGA Championship at Whistling Straits in Wisconsin, where he will be chasing his third Major of the season.
Korean-born leader Lee, meanwhile, took sole possession of the lead after sinking a succession of long putts to shoot 65.
Teeing off on the back nine, Lee opened with a pair of birdies on the 10th and 11th before dropping a shot on the 12th with a bogey after his tee shot found the left greenside bunker.
However, Lee came roaring back on the 13th, draining a 47-foot putt to move back to two under after four holes.
A bogey on the par-three 15th was offset by a further birdie on the 16th, when an iron to the green left him with a two-foot putt.
Three more birdies at the start of the front nine, including a 29-foot putt on the par-five second and a 28-footer on the fourth, took Lee to five under.
"It was a good day for me," said Lee. "I left couple short birdie putts out there. It may sound a little bit too greedy, but I did. And I was hitting it that good, and I was very happy with my result."
One off Lee's pace were American veteran Jim Furyk and Northern Ireland's Graeme McDowell, who both carded 66s.
Furyk had appeared set to grab a share of the lead only to come unstuck with a bogey on his final hole.
McDowell finished on four under after a round comprising five birdies and one bogey, leaving the European Ryder Cup star well placed to snap a miserable run of form which has seen him post only two top-50 finishes this year.
Rickie Fowler and England's Justin Rose were a further shot back on three under after carding 67s, one ahead of a group of eight players tied for sixth on two under that included Lee Westwood, Ian Poulter and Paul Casey.
Newly minted British Open champion Zach Johnson, meanwhile, was back with Spieth in a group of 15 players at level par.
Leading scores from the first round of the $9.25m WGC-Bridgestone Invitational at Firestone Country Club on Thursday (USA unless stated):
65 - Danny Lee (NZL)
66 - Graeme McDowell (NIR), Jim Furyk
67 - Rickie Fowler, Justin Rose (ENG)
68 - Victor Dubuisson (FRA), Paul Casey (ENG), Robert Streb, Koumei Oda (JPN), Lee Westwood (ENG), Ian Poulter (ENG), Webb Simpson, Henrik Stenson (SWE)
69 - Soren Kjeldsen (DEN), Francesco Molinari (ITA), Jason Day (AUS), Dustin Johnson, Keegan Bradley, Kevin Na, Brooks Koepka, Hideki Matsuyama (JPN)
70 - James Hahn, Jamie Donaldson (WAL), Shane Lowry (IRL), Danny Willett (ENG), Thomas Bjorn (DEN), Charl Schwartzel (RSA), Brandt Snedeker, David Lingmerth (SWE), Bubba Watson, Marcel Siem (GER), Martin Kaymer (GER), Joost Luiten (NED), Jordan Spieth, Zach Johnson, Russell Henley
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