World number two Spieth, seeking a career Grand Slam at age 24 after winning the British Open last month.
World number two Spieth, seeking a career Grand Slam at age 24 after winning the British Open last month, struggled to a one-over 72 in the opening round at Quail Hollow, five shots behind Danish leader Thorbjorn Olesen.
Jordan needed 32 putts to navigate the faster-than-expected greens, which proved formidable despite absorbing an inch of rain at the start of the week.
"I can't putt any worse than I did today," Spieth said. "If you told me I was going to hit my driver the way I did today, I would have definitely thought I shot a few under par, which was an awesome score.
"The score won't be any higher than it was today if I'm driving the ball like today."
The disappointing score came as Phelps, who set a single Olympics record with eight gold medals at Beijing in 2008, walked the course watching Spieth launch his bid to become only the sixth man to win all four major titles in his career, and be the youngest to do it.
"It was great," Spieth said. "He has become a good friend and mentor. It's pretty awesome to have a mentor like that. He texted earlier in the week. I think he was here for an outing and said he was going out to be out following today."
"We've had dinner, talked through a lot of things that I will probably just keep to myself. A lot of mental approach and preparation stuff."
Such things are just what Spieth will need if he is to lift the Wanamaker Trophy on Sunday and join Tiger Woods, Jack Nicklaus, Gene Sarazen, Gary Player and Ben Hogan on the career Slam list.
"When I had the chances that I had and I just couldn't get the ball to go in on the greens, that is when I get the most frustrated I can get out there," Spieth said.
"I don't think I missed any short putts. I just had really poor speed on my really long ones. Everything else was fine. It was just the putter."
Spieth began on the back nine and made the turn at level par but had three bogeys in the first six holes on the front, missing 10-foot par putts in each case.
A three-footer for birdie at the par-5 seventh and a tap-in birdie at eight lifted some of the gloom.
"If I were to finish par, par, par, I would have thrown myself out of the tournament," Spieth said. "I had three looks (at birdies) and almost got back all the way to even.
"I needed those from short range because it seemed like the lid was on today. I burned a lot of edges. Some were good putts, some weren't. Overall that's what I'm going to work on is hitting some more putts."
The late rally gave Spieth hope his victory dream can still happen, but he knows it puts more pressure upon Friday's round.
"I'm still in it but I know tomorrow's round becomes that much more important to work my way and stay in it," Spieth said.
"I've got to make up ground. If I'm five back at the start of the day, I've got to be less than five back after Friday to really feel like I can play the way this golf course needs to be played and still be able to win."