Title-chasing Warwickshire still have a good chance of winning their first County Championship in nine years but will have to bowl out Somerset on the final day at Edgbaston.
The Bears closed on 179-1, with a 157-run lead, needing quick runs to stand a chance of forcing victory by declaring and setting Somerset a tempting target.
Rob Yates is unbeaten on 72 with Will Rhodes on 42 after Dom Sibley made 50.
Somerset, who made 389, earlier denied the Bears maximum bowling points.
That would have proved crucial if Hampshire had beaten Lancashire in Liverpool – but the Red Rose hung on to win by one wicket in a nail-biting finale at Aigburth.
That put the Bears’ hopes back in their own hands and if they can close out victory over Somerset on Friday, Warwickshire will earn an eighth county title – and their first since 2012.
England all-rounder Chris Woakes is the only survivor from that team, having made seven Championship appearances in 2012, and he did his utmost in a taxing morning session to try to enforce another title.
In only his second Championship appearance this summer – after a three-year absence from the Bears’ red-ball team – he bowled tirelessly with no luck.
Woakes (3-100 from 30.3 overs), Tim Bresnan (3-35) and spinner Danny Briggs (3-77) took the lion’s share of the wickets.
But Warwickshire failed to get that key ninth wicket inside the 110-over cut-off mark as two other England internationals, Lewis Gregory (60) and Jack Leach (49), led the Somerset resistance.
After some dismal batting failures since the Championship restart, Somerset made 389, their highest score since mid-July, to even earn a first-innings lead of 22.
Yates and Sibley soon ate into that to complete Warwickshire’s first century opening partnership of the season before Rhodes raced to an unbeaten 42 off 29 balls – with the prospect of more quick runs on Friday morning from the likes of Sam Hain, Woakes, Bresnan and Briggs.
“What’s happened at Liverpool has put it back in our hands so now we have got to try and set something up tomorrow and take 10 wickets.
“Fair play to Somerset. They came out and did what they needed to do and took a lot of time out of the game. But the last session has swung it back in our favour a little bit and we are going to have to play well in the morning to come up with a target with a suitable amount of overs that we think we can bowl them out in.
“When you have had 13 and three-quarters games and are still in the hunt for a County Championship, that’s a great feeling. There is a lot of excitement in the dressing room at the minute. We just know that we have all got to make it as hard as possible for Somerset.”
Somerset all-rounder Lewis Gregory:
“Batting is difficult against the new ball and Warwickshire bowled well in the first hour, but we batted well and put some good partnerships together.
“Whether we chase a target is whether it is realistic. We always want to win games of cricket, but the chase has to be realistic.
“Warwickshire need to give themselves time to bowl us out.”