The Aussie, who is bidding to become the world number one for the very first time, shot a 67 to sit just two behind the current leader.
Smith has also dealt with a number of questions about his future, and has refused to deny intense speculation he is joining the Saudi-backed breakaway LIV Golf at the end of the season.
Spaun maintained his place at the top of the leaderboard for the third round in a row after a couple of late birdies retained his advantage at 13 under.
He leads by one from Austria's Sepp Straker, but there is a growing threat from the group in joint-third where Smith was joined by Will Zalatoris - runner-up at the US PGA and US Open this year - after an impressive 65 and Trey Mullinax, who shot 66.
But all eyes will be on Smith on Sunday as he seeks to reach another landmark.
"That would mean a lot. I mean, that's what we're all here to do," said Smith on the possibility of overtaking Scottie Scheffler at the top of the rankings.
"One of my goals, probably since the start of the year, is to try to get to that top spot and I'll try to chase it down."
Both Spaun, world 98 and Valero Texas Open champion and Straka, February's Honda Classic winner, are seeking to become the ninth multiple winner on the PGA Tour this season.
US Open champion Matt Fitzpatrick remains in contention four shots back after a 67 lifted him to nine under and into a share of eighth place.
Victory in Memphis guarantees the winner a place East Lake for the season-ending Tour Championship, where an 18million US dollars (£14.9million) bonus is up for grabs for the winner of the FedEx Cup.