A third-round 67 leaves the Bathgate man fourth, three shots off the lead shared by England’s Callum Shinkwin and American Julian Suri.
Overnight leader Suri was two strokes clear of Shinkwin before the latter finished his round with a double-bogey seven at the 18th.
Posting 71, it left him on 11 under alongside Shinkwin, who had gone round in 68, and two shots clear of third-placed Masahiro Kawamura.
Japan’s Kawamura is on nine under, with Gallacher one shot behind him and Swede Robert Karlsson another stroke back in fifth.
Gallacher, a three-time winner on the European Tour, shot a five-under-par round which included seven birdies and two bogeys.
Shinkwin, playing on an invitation in New Delhi after finishing 119th in last season’s Race to Dubai rankings, is bidding for a first European Tour title.
After registering five birdies and one bogey on Saturday, the 25-year-old told europeantour.com: “I lost my card last year by a couple of spots so I know I can get it back if I play well so I just have to trust what I can do and let it happen.
“I just want to play as well as I can. If I do win that’s great, if I don’t then I know my time will come soon.”
Suri, having started the day two shots clear at the top, opened with birdies at the first and second to briefly increase his lead to four.
That was down to one after dropping shots at the third and seventh, but after birdieing three holes in a row from the eighth to the 10 it was back up to three.
He subsequently recovered from bogeying the 14th with a birdie on the next hole, before concluding the round in frustrating fashion.
Suri, who is playing just his second event of the season after surgery on an abdominal hernia, said: “I think a couple of the mistakes I made today were after long waits on the tee box and that was quite a long wait on 18.
“No excuses, totally my fault, and I need to compose myself better. I think that is a bit of a learning curve for me after being out of tournament golf for a while now.
“I am feeling good with my overall game despite 18 there. The game feels good.”