Newcastle and Leeds Carnegie are one win away from completing an English nap-hand in this season's European Challenge Cup.
Bath's 31-13 success against Overmach Parma means they joined Premiership colleagues Sale Sharks and Worcester in the quarter-finals.
Newcastle will follow suit by seeing off Spanish minnows Cetransa El Salvador at Kingston Park next Sunday.
But Leeds' task is infinitely tougher, knowing they need victory over French hosts Castres on Thursday night to guarantee progression as a group winner.
Bath celebrated head coach Steve Meehan agreeing a new two-year contract by recording a bonus point success in Italy.
Matt Banahan, Bath's 6ft 7in wing, claimed a try double, and there were also touchdowns for fly-half Butch James and flanker Michael Lipman, with James kicking one conversion and Olly Barkley booting nine points.
Castres kept the pressure on Leeds in Pool Five through an impressive 27-9 success away to Calvisano.
Back-row forwards Lei Tomiki and Joe Tekori, together with Uruguayan international Rodrigo Capo Ortega, scored tries for the visitors.
Auch kept their faint hopes of a quarter-final alive with a 15-13 win over Albi.
Auch trail Pool One leaders Bath by an unreachable nine points, but have a slim chance of reaching the last-eight as one of the three best runners-up.
They need to upset Bath next weekend, and then hope several other results go their way.
Flanker Nicolas Bontinck and centre Pierre Aguillon scored tries as Auch took a 12-5 lead into the interval, and they held on for the win, despite an Albi comeback inspired by centre Dimitri Senio and Patrice Serre.
In Pool Four, Montpellier joined the queue for a best runners-up spot with a nervy 26-24 win at Bayonne.
The victory - thanks mainly to four penalties and two conversions from fly-half Federico Todeschini - saw Montpellier home, although Bayonne collected two bonus points after scoring four tries through Jean-Baptiste Peyras, Sam Gerber, Arnaud Heguy and Jean-Jo Marmouyet.
As head coach Stuart Lancaster polishes the axe he will soon wield over his World Cup training squad, Exeter Chiefs coach Rob Baxter has backed Jack Nowell, Henry Slade and Luke Cowan-Dickie to be given a stay of execution.
Italy Under-20s held off a late Samoa blitz to retain their place in the World Rugby U20 Championship on Saturday and head coach Alessandro Troncon believes the experience will have served as a valuable lesson for his young troops.