Gloucester's home game against Bayonne next month will decide Pool Three, while the other four qualifying groups are also currently topped by English clubs.
London Irish staged a thrilling fightback to defeat Pool Two rivals Agen 32-24 at Stade Armandie, a result that opened up a nine-point gap between the Exiles and their French challengers.
Agen led 24-12 at one stage of the second-half but Barry Everitt chipped away at the home side's advantage with two penalties before Irish scored two converted tries during injury time.
Australian centre Paul Franze, a summer recruit from rugby league side Penrith Panthers, claimed the first touchdown, then full-back Delon Armitage raced over for an interception try to leave unbeaten Irish one more win away from the quarter-finals.
In the other Pool Two fixture, Italian outfit Parma defeated French strugglers Pau 20-11 thanks to some high-class goalkicking from full-back Ezio Galon.
Worcester also experienced a dramatic finale in their Pool Five clash with Connacht in Galway but they were on the receiving end as flanker Matt Lacey's 88th-minute try denied the Warriors 22-21.
Worcester though, having beaten Connacht at home last week, will still top the group if they claim bonus-point victories from their remaining two games against Montpellier and Catania.
Hooker Chris Fortey rumbled over for the Warriors in the first-half, with Shane Drahm's boot giving Worcester an 11-10 interval lead.
South African Thinus Delport extended Worcester's advantage by scoring a try before Lacey came up trumps and left the Warriors contemplating a tricky appointment with Montpellier in France next month.
Borders' quarter-final hopes nose-dived after they were edged out by Pool Four hosts Brive 34-28, a result that guarantees Newcastle a quarter-final spot if they collect nine points from their last three games, two of which are at Kingston Park.