Mitchell Starc shines an England collapse at Lord's on Saturday before Ben Duckett was given a controversial reprieve as the hosts maintained their bid for a remarkable win in the second Ashes Test.
England were 114-4 at stumps on the fourth day, still needing a further 257 runs to reach a target of 371.
If achieved, that would be England's second-highest run-chase to win a Test behind their 378 against India at Edgbaston last year.
Left-handed opener Duckett, who made 98 in the first innings, was 50 not out and captain Ben Stokes unbeaten on 29, with England looking to level the five-match series at 1-1.
Starc, recalled following Australia's thrilling two-wicket win in last week's first Test at Edgbaston, reduced England to 13-2.
And they were in dire straits at 45-4 when Australia captain Pat Cummins struck twice in an over.
Duckett then appeared to fall for exactly 50 when he miscued Cameron Green to fine leg, where a sliding Starc held a fine catch only for the ball to touch the turf.
Third umpire Marais Erasmus, to Australia's visible despair, disallowed the dismissal because Starc had grounded the before he was in control of his body and so had not completed a fair catch.
"I have no idea on the rules, we've seen different ones in this game and different decisions have been given," England batting coach Marcus Trescothick told the BBC.
Bd-pratidin English/Lutful Hoque
