Rose, who became the first English winner of the US Open for 43 years at Merion in 2013, carded a third round of 67 to finish six under par, a total matched by Ryder Cup team-mate Sergio Garcia.
Garcia's fellow halfway leader Rickie Fowler was a shot behind, with Spieth another stroke back alongside compatriots Ryan Moore and Charley Hoffman.
Former champions Adam Scott and Charl Schwartzel were three and four shots off the pace respectively, with Lee Westwood and Thomas Pieters on one under.
World number two Rory McIlroy, who needs to win to complete the career grand slam, and Paul Casey were part of a six-strong group on level par.
Rose was one over for the day after seven holes, but birdied the eighth and picked up five more shots in the last seven holes in a back nine of 31.
"This is a place I dearly love and would love to be part of the history here," said the 36-year-old, who was joint second behind Spieth in 2015. "Tomorrow is a huge day. I have an opportunity and that's all that you want, but it all starts on the back nine.
"It all happened on the last eight holes today and the patience I showed on the front nine was probably key on the back nine. I had my head down and knew I was creeping up on the leaders.
"I've started to hit quality iron shots to set up some opportunities and today I felt really comfortable with the putter, which was great. The finish was a bonus, hitting it close on 17 and on 18 to get it on the right level.
"I've been trending (in the right direction) all year, playing some good golf. I've been waiting for an extra gear, I seem to be finding it and hopefully can keep trending the way I am.
"I take confidence from playing in the final group with Jordan in 2015 and only losing to a record performance. Many other years my score would have been good enough to win."
Defending champion Spieth held a five-shot lead with nine holes to play in 2016 and looked certain to become only the fourth player, after Jack Nicklaus, Nick Faldo and Tiger Woods, to win back-to-back Masters titles.
However, after dropping shots on the 10th and 11th, Spieth hit two balls into Rae's Creek on the par-three 12th and ran up a quadruple-bogey seven, with Danny Willett taking full advantage with a closing 67 to win.
The 23-year-old safely negotiated the 12th on his return 12 months later, only to then stumble to a quadruple bogey on the 15th in an opening 75 which left him 10 shots behind Hoffman.
Harry Vardon in the 1898 Open is the only major champion to have been 10 or more shots behind after the first round, but Spieth carded a 69 on Friday and Saturday's 68 included a tap-in birdie on the 15th after an approach "with the same wedge, same number" which span back into the water on Thursday.
"Disaster avoided," Spieth said with a smile. "I'm going to be behind but I'm in a great position to make some noise early on and hopefully hold it up.
"After contending on Sunday for a few years in a row you normally don't need as low as you think. Something like today's round and maybe finish it off a little better and it might be enough."
Spieth led from start to finish in 2015 and for the first three rounds in 2016 and added: "It will be nice for a change, fun to have a new experience coming from behind."
Garcia, who has recorded 22 top-10 finishes in majors without success, has the added incentive of being able to claim a first title on Sunday, which would have been the 60th birthday of two-time champion and fellow Spaniard Seve Ballesteros.
His 70 was a massive improvement on his third round scoring average of 74.92 and featured a back nine of 34 thanks to birdies on the 13th and 15th.