Surrey 239 for 8 (Stoneman 67, Burns 50, Leach 5-42) trail Somerset 429 (Clark 3-75) by 190 runs
Jack Leach took five wickets as Somerset celebrated securing their place in the first division by putting Surrey under pressure on the third day at the Kia Oval.
The England left-arm spinner finished with 5 for 42 from 33 including 19 maidens as Surrey reached 239 for 8 in reply to Somerset's 429.
Hampshire's failure to secure maximum batting points against Gloucestershire ensured Somerset's place in the top-flight when the LV= Insurance County Championship concludes in September, and Leach bowled outstandingly to give them an outside chance of finishing Group 2 with a victory.
It was their other slow left-armer, Roelof van der Merwe, who made the breakthrough when Burns top-edged a sweep shortly after reaching 50 which included a reverse-sweep for six off Leach and six fours.
It was about the only occasion when Leach was collared as settled into a good rhythm, providing control and getting the ball to turn on a slow surface. In six previous Championship games this season he bowled 130 overs in total and clearly relished three long spells here.
Stoneman had played nicely for his 67, with ten fours, when he was deceived by Leach's arm ball with the total on 136 before Leach picked up the key wicket of Hashim Amla. The South African faced 278 balls for an unbeaten 37 against Hampshire last week and was settling in for another long vigil when he was caught and bowled off a leading edge for 16 from 66 balls.
Leach switched to the pavilion end to snap up Jamie Smith who was caught behind off one that turned just before tea. Immediately after the resumption Ravi Ashwin edged his first ball to second slip and Leach had his five-for when Devon Conway ran back from mid-wicket to take a great catch over his shoulder off Ryan Patel's top-edged sweep.
Somerset delayed taking the new ball, preferring to let Leach and van der Merwe probe away, but when they did Jordan Clark and Rikki Clarke batted sensibly to put on 42 in 21 overs. Clark had made 33 when van der Merwe, who gave Leach good support with 3 for 54, returned and had him caught at second slip before bowling Jamie Overton around his legs in the last over.