Northamptonshire 105 and 56 for 3 need 264 more runs to beat Kent 137 and 287 (Denly 81, Crawley 63, Billings 56, Gleeson 6-79)
Kent will need to mop up Northamptonshire's seven remaining wickets on the final day at Canterbury to secure their eighth win of the Specsavers County Championship campaign having reduced the visitors to 54 for 3 by stumps on day three.
Batting for the final time in the match and facing an improbable victory target of 320, Northamptonshire started their second innings as they had their first and were soon deep in trouble against the pace of Matt Henry, who has already taken nine wickets in the match - including a career-best seven-wicket haul.
Left-hander Ben Curran followed one going across him to be caught behind then, five overs later, his opening partner Luke Procter went leg before to one that darted in off the seam to give Henry his ninth of the match.
Kent's first-change seamer Grant Stewart swung one through the gate of Ricardo Vasconcelos to peg back off stump, but Alex Wakely and Richard Levi dug in to bat out the remaining 10 overs through to the close. They require a further 266 on the fourth day for an unlikely win.
Kent had started their second innings with a slender overall lead of 32 and extended their advantage to 55 when a double-wicket maiden from Ben Sanderson briefly altered the course of the morning session.
He had Sean Dickson snaffled at second slip then, three balls later, Kent's experiment to promote Grant Stewart backfired, when the allrounder glanced a catch to the keeper to depart without scoring.
With licence to keep the game moving, Kent's third-wicket partners Zak Crawley and Joe Denly continued to go for their strokes in adding 111 in 25 overs either side of lunch.
But, after posting a 72-ball 50, Crawley became bogged down after the interval and departed leg before for 63 when an in-swinging Brett Hutton yorker hit him flush on the boot.
The impressive Richard Gleeson then enjoyed the day's second double-wicket maiden. After clipping Denly's off stump with an excellent leg-cutter that sent him packing when only 19 runs short of his fourth championship ton of the summer, Gleeson extended Daniel Bell-Drummond's batting woes by trapping him leg before when off balance for a third-ball duck.
Heino Kuhn fell for a cameo 34 just before tea, leg before when working across the line against Gleeson, then, soon after tea and with Kent 287 ahead, the same bowler induced Darren Stevens to chop on.
Harry Podmore followed suit to give Gleeson a deserved five-wicket haul in only his fourth appearance of the season, which became a career-best of 6 for 79 when Henry, in trying to drop his hands to a short one, gloved a catch through to the keeper.
Amongst the carnage, Sam Billings, the Kent captain, scored his first half-century of the season in his fifth appearance from 63 balls and with seven fours, beating his paltry season's best of 29 before becoming last man out for 56, after an extraordinary hook from well outside off stump that flew straight to first slip.
Kent needed only 25 minutes at the start of the day to mop of Northamptonshire's two remaining first innings wickets.
Henry had Richard Gleeson caught by Ivan Thomas at square leg to end a ninth-wicket stand of 49 to give Henry career-best figures of 7 for 42. Thomas then polished off the innings in Henry's next over, running out Hutton with a direct hit from square leg, allowing Henry to walk off with the match ball.