The defending champions are 72 runs ahead needing 4 wickets for victory, and could still bat again.
Leaders Nottinghamshire trail Durham by 190 runs with 9 wickets in hand, after the hosts posted 664 in their first innings at Chester-le-Street yesterday.
Hampshire beat second-placed Warwickshire by 89 runs, and Somerset won by 5 wickets in their match with Sussex.
Bottom side Worcestershire picked up their first victory of the season, easing to a 225 run victory over Essex.
