The American had two more holes to complete and trailed Song by three strokes at the Sentosa Golf Club when the horn sounded for the remaining flights to return to the clubhouse.
Three other players are also hot on the heels of the South Korean in the $1.0 million tournament.
China's Liang Wen-Chong trails by a stroke, and Japan's Shintaro Kobayashi and South African Keith Horne by two.
The last flight that included Song completed only 13 holes, and a total of 15 players will return to the Serapong Course on Sunday morning to resume play before the start of the final round.
Spieth, 22, had a mixed day, playing the remaining 12 holes that were carried over from Friday after play was halted due to inclement weather.
The Texan finished the round with three birdies and two bogeys.
“The bogey did hurt on the last,” said the US Open and Masters champion before heading for the penultimate round.
Spieth is playing without regular caddie Michael Greller, who is recovering from an ankle injury sustained at the Hyundai Tournament of Champions in Hawaii earlier this month.
His agent, Jay Danzi, is standing in as Greller's replacement.
“I made all my putts inside six feet yesterday but missed five or six today. I've just got to get comfortable on the shorter length putts,” said Spieth, who is playing in Singapore for the first time.
“These kinds of delays (due to bad weather) have happened before, in professional and college golf, you have to get used to it. Gotta be prepared to play a lot of golf.”
Spieth's erratic form continued in round three.
Despite two birdies in the first nine, he struggled to find the fairway from the tees, hit the bunker four times and failed to make par on the fifth hole.
Although he was hit by another bogey in the home stretch, Speith's form kicked in.
His drives found their mark and the short game found the putts, hitting another birdie on the 12th hole.
Song failed to make any headway, after putting in an outstanding round the previous day, with bogeys in the fifth and 11th holes cancelling out the two birdies he shot on the third and fourth.
But he remains hopeful of snatching a first career win on Sunday despite the chasing pack.
“I have had many second place finishes in Japan and Korea but I do not think that will affect me (mentally) tomorrow,” said the 24-year-old.
Completed scores after round three:
205 - Liang Wen-chong (CHN)
207- Wang Jeunghun (KOR), Lee Won-joon (AUS), Akio Sadakata (JPN)
208 - Berry Henson (USA), Choi Jinho (KOR), Brett Munson (USA)
209 - Sam Brazel (AUS)
210 - Daisuke Kataoka (JPN), Lee Soomin (KOR), Masahiro Kawamura (JPN), Yoshinori Fujimoto (JPN), Chapchai Nirat (THA), Yuta Ikeda (JPN), Yusaku Miyazato (JPN) – AFP
Original source: Spieth in the hunt in Singapore