But the visitors head into the fourth and last day of action knowing collecting eight wickets will see them to victory.
Earlier in the day Hampshire were all out for 322 in their second innings, with the ninth-wicket pair of Keith Barker and Brad Wheal having added 93, writes Brian Halford.
Home openers Rob Yates (70 not out) and Dom Sibley (47) added 99 in 50 overs to give Warwickshire the perfect platform as they chased 296 for victory.
But England international spinner Liam Dawson then struck twice to keep the visitors very much in the match.
The Bears, who will remain top of the table if they win, still have plenty to do although Hampshire, already without the injured Kyle Abbott, suffered another blow when pace spearhead Mohammad Abbas limped off just before tea.
After resuming on 260-8, former Edgbaston favourite Barker struck 75 off 192 balls and Wheal - striking a career-best 45 - stretched their stand to 42 overs before the former lifted Chemar Holder to Chris Benjamin at point.
On a flattening pitch, ex-Hampshire spinner Danny Briggs then rounded off the innings by having Abbas caught close in by Yates to leave Warwickshire a target of 296 in five sessions and two overs.
Openers Yates and Sibley negotiated the potentially awkward spell before lunch and then stayed together for 33 overs between lunch and tea.
They concentrated hard and left skilfully to post a half-century partnership in 29 overs.
Deprived of Pakistani seamer Abbas from mid-afternoon, Hampshire's attack persevered impressively to ensure that, although wickets were not falling, runs arrived slowly enough to keep them in the game.
The opening stand was worth 99 when Sibley skied an attempted sweep off Dawson and wicket-keeper Tom Alsop accepted the simplest of catches beside the stumps.
Dawson then quickly added the wicket of Chris Benjamin, who tried to increase the scoring rate, but perished in the attempt when he was caught by Barker at mid-on.
Yates (70*) and Sam Hain (five not out) defended through the last nine overs to the close and a match which has been finely balanced throughout remains so heading into the final day.
Hampshire all-rounder Wheal said: ‘I think it's going to be a good day of cricket ahead. We are in a decent position and a couple more wickets early doors in the morning and we put ourselves in a very good spot to win this game.
‘The pitch is getting a little bit slower and flatter as the game goes on, but there is definitely enough for the bowlers. We are just trying to keep it tight, they are just going at about two runs an over, and hopefully the chances will come.
‘I was pleased with my innings. Barks (Keith Barker) batted beautifully and he was a good partner to have throughout that stand. We were just saying we want to get as many runs on the board as we can and bat as long as we can so that we can put their guys under pressure and give ours something to bowl at so it was good fun.’