Saracens will qualify for the Heineken Cup quarter-finals if they beat Connacht next Saturday - but a prized home tie looks to have eluded them.
The Aviva Premiership leaders must keep their side of the bargain, then hope that Italian minnows Zebre pull off a monumental shock by defeating four-time European champions Toulouse.
Saracens do look set to fill one of the two best runners-up spots, though, and it is possible they could face a return trip to Toulouse in the last eight.
A 21-11 defeat against Toulouse at Stade Ernest Wallon put the French side firmly in control of Pool Three.
"Your set-piece has to be good to gain momentum, and although our scrum was good our lineout didn't function," Saracens rugby director Mark McCall told the club's official website after the loss.
"But it has happened and we have six days to turn things around for Connacht, which is a huge game.
"If we can win that game we get the chance to go somewhere else in the quarter-finals and prove we are better than we were (on Sunday).
"Obviously, we are very disappointed. We just don't think that we gave the best account of Saracens. We just couldn't get any momentum in our play."
Northampton's hopes of increasing the English quarter-final contingent - Leicester are already through - appear remote despite their 29-17 Liberty Stadium victory over Pool One hosts the Ospreys.
The Saints need a bonus point win against Franklin's Gardens visitors Castres next Friday night, and have to hope that group leaders Leinster collect nothing from a Dublin appointment with the Ospreys.
Northampton could still progress as one of two best runners-up if they topple Castres, but that is an improbable scenario, given the remaining fixtures in other groups.
"From our point of view, we've got one game left and we are going to go flat out and try to get five points," said Saints' Wales wing George North, who claimed a 70-metre breakaway try and was joined on the scoresheet by George Pisi and Glenn Dickson.
"It has been a tough group for us to get back into after a couple of early losses, and it is now win or nothing for us against Castres. We've still got a chance of making it, and we will go hard until the end."
Forwards coach Jono Gibbes praised Leinster's resilience as they came from two converted tries down to beat Castres 29-22 at Stade Pierre-Antoine.
Fly-half Jimmy Gopperth inspired Leinster's comeback, scoring two first-half tries in his 21-point haul, with man-of-the-match Rob Kearney chipping in with a drop-goal and Jordi Murphy adding a clinching late try.
"It took a massive effort from us to score three tries, and we almost sniffed out an opportunity for a fourth at the end," Gibbes said.
"To come to a place like this against a well-organised team that haven't given up a lot of tries at home, it's a satisfying result and puts a spring in our step for next Friday."
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