Shinkwin and Suri, with a total score of 11-under 205, are two shots clear of Japan's Masahiro Kawamura who managed to shoot 68 on the DLF course in Gurgaon, near Delhi.
Scotland's Stephen Gallacher, hitting three-under, and Robert Karlsson of Sweden at 4th and 5th also kept themselves in the hunt for the $1.75-million event - co-sanctioned by the Asian and European Tour.
But the day belonged to Shinkwin who will be gunning for his maiden European Tour title in the final round after missing his Tour card last season.
"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," he said.
"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, who held the sole lead overnight, hit a double bogey in the 18th hole and Callum played catch up.
S. Chikkarangappa carded six-under to move into tied 6th and keep India's hopes alive as home favourite Shubhankar Sharma, a two-time European Tour winner, improved to tied 15th.
Chikkarangappa (74-70-66) revived his career last year with a runner-up finish at an Asian Tour event that helped him seal his card in Asia for 2019.
"I will take seven birdies and a bogey on this golf course any day. This is my best round at this course and almost feels like a course record," said Chikkarangappa.
"As for the last round, I just need to enjoy myself on the course and play with no expectations. Hopefully, the rest should fall in place."
Leading third-round scores from the Indian Open at the par-72 DLF course in Gurgaon on Saturday:
205 - Callum Shinkwin (ENG) 72-65-68, Julian Suri (USA) 67-67-71
207 - Masahiro Kawamura (JPN) 69-70-68
208 - Stephen Gallacher (SCO) 67-74-67
209 - Robert Karlsson (SWE) 68-68-73
210 - S. Chikkarangappa (IND) 74-70-66, James Morrison (ENG) 70-71-69, Matthias Schwab (AUT) 72-68-70, Mikko Korhonen (FIN) 71-69-70, George Coetzee (RSA) 70-66-74
211 - Sihwan Kim (USA) 76-68-67, Erik Van Rooyen (RSA) 69-73-69, Scott Hend (AUS) 69-68-74
212 - Jens Dantorp (SWE) 70-72-70
213 - Victor Dubuisson (FRA) 71-72-70, Shubhankar Sharma (IND) 69-73-71, Ashley Chesters (ENG) 69-73-71, Prom Meesawat (THA) 68-74-71, Lorenzo Gagli (ITA) 76-65-72, Nacho Elvira (ESP) 72-69-72