South Africa were 160 for three in their second innings, a lead of 290 runs, at lunch on the third day.
Amla, who made 78 in the first innings, was 61 not out and returning captain Faf du Plessis four not out.
Together with opener Dean Elgar (80), Amla put on 135 for the second wicket.
England will need to set a new record if they are to achieve an improbable victory, with the most any side have made in the fourth innings to win in 118 years of Test cricket at Trent Bridge their very own 284 for six against New Zealand in 2004.
Elgar was 38 not out and Amla 23 not out.
James Anderson, who took five for 72 in South Africa's first innings 335 -- the seventh Test five-for recorded by England's all-time leading wicket-taker at Trent Bridge -- beat Elgar's outside edge.
But when Stuart Broad, bowling on his Nottinghamshire home ground, challenged Elgar from around the wicket, the left-hander responded with an off-driven four.
Broad produced an excellent delivery to Amla, then on 25, which carried to wicket-keeper Jonny Bairstow.
England appealed half-heartedly and then decided against reviewing Australian umpire Simon Fry's not out decision, only for replays to indicate there had been an edge.
England needed every possible chance to stick in their dire position.
Elgar, however, had a reprieve on 55 when a thick edge off Broad flew high to gully where Anderson, diving to his left, so nearly held what would have been a spectacular catch.
The ensuing confusion saw Ben Stokes miss with a shy at the stumps.
England eventually had a breakthrough when the painstaking Elgar fell in sight of would have been an eighth Test century when he took his eye off a well-directed Stokes bouncer and tamely mis-hooked to Anderson at backward square leg.
South Africa's 153 for two soon became 154 for three when Quinton de Kock was caught behind off Anderson.
But Du Plessis, who didn't play at Lord's after the birth of his first child, got off the mark with a legside boundary against Anderson that left England needing to rewrite the record books.
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