Rose carded a second round of 67 at East Lake to finish seven under par, a total matched in the following group by 14-time major winner Woods, who added a 68 to his opening 65.
A second win of the season for Rose on Sunday would see the Englishman secure the overall FedEx Cup title and a bonus of £7.6million, while Woods is seeking a first victory since 2013 to put the seal on his remarkable comeback following spinal fusion surgery in April 2017.
Woods covered the front nine in level par despite finding just two fairways, but birdies at the 12th, 14th and 15th briefly gave him a two-shot lead before a costly double bogey on the 16th, where he was forced to play out sideways from a fairway bunker after pulling his tee shot into thick rough.
The 42-year-old bounced back to birdie the par-five 18th and join Rose at the top of the leaderboard, with Rory McIlroy two shots off the lead on five under and Jon Rahm, Justin Thomas, Patrick Cantlay and Billy Horschel a shot further back.
Rose, who carded five birdies and two bogeys in his 67, told Sky Sports: "(I'm) happy with that score, it was a little rough around the edges compared to yesterday but you can't play like yesterday every single day. Happy that I was able to find a score out there for sure.
"I was waiting for a bit of momentum, felt like the round had got a little flat around the turn and then those two birdies in a row (on 13 and 14) really got me going. I took a look at the leaderboard at that point and realised that no-one had gone anywhere today, it was obviously a bit trickier for some reason."
As for playing with Woods in the final group on Saturday, Rose added: "I'm very much looking forward to it, in my mind he's the greatest of all time just cause he's in the era I've witnessed.
"It will be an honour to play with him as world number one but there's more at stake than playing with Tiger Woods and I'm looking forward to another day of business and edging my way towards an exciting Sunday."
Woods, whose last win came in the 2013 WGC Bridgestone Invitational, told Sky Sports: "I ground out a round today, I wasn't as sharp as I was yesterday and missed a few shots but hung in there with a bunch of pars and made a couple of putts which was nice."
McIlroy, who won the Tour Championship and FedEx Cup title in 2016, was left to rue failing to take advantage of the par fives as he added a 68 to his opening 67.
"I'm making enough birdies but I didn't birdie either of the par fives which is disappointing," the four-time major winner said. "If I had been able to do that I might have been a little closer to the lead but no-one is going anywhere out there it seems.
"I'll probably be one or two behind and it was nice to play the last four holes in two under par."