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The previous highest chase by England at Lord's is 282-3 against New Zealand in 2004, meaning Alastair Cook's men are likely to face a record target.
Asad Shafiq (49) and Sarfraz Ahmed (45) batted Pakistan into a strong position.
England bowled well - Chris Woakes taking another five-wicket haul - but dropped two catches late in the day.
Stuart Broad spilled a difficult chance to dismiss Yasir Shah (30 not out) at mid-off and then wicketkeeper Jonny Bairstow dropped what was more of a regulation catch to end a punchy innings from strokemaker Sarfraz.
They may also rue the moment Younus Khan, who put on 69 with Shafiq for the fifth wicket, had a leg-before dismissal reversed following a review with the score at 75-4 and Pakistan's lead just 142.
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Woakes was head and shoulders above the rest of his team-mates with the ball.
The Warwickshire all-rounder started the day hoping to push England closer to Pakistan's first innings total of 339 all out.
However, he was let down by his batting partners as England's last three wickets fell for 19 runs - leaving Woakes unbeaten on a well-constructed 35.
He then shone with the ball, recording what are currently the second-best match figures (11-101) by an England bowler this decade.
His expert line and length, which is delivered at a livelier pace than when he made his debut in 2013, saw him snare opener Shan Masood via a catch at first slip and helped him to pin Azhar Ali in front.
He returned later in the day to bowl the obdurate Asad Shafiq for 49 and have both Sarfraz and Wahab Riaz caught behind with an outswinger and attempted bouncer respectively.
England's batsman were bamboozled by the leg-spin of Yasir Shah in their first innings, with Steven Finn becoming the 30-year-old's sixth victim on a frenetic third morning.
The odd delivery is beginning to stay low, while Woakes showed the benefit of a lively line and length on a sporting surface.
History is also stacked against Alastair Cook's side. Their highest chase at Lord's is the 282-3 they scored against New Zealand in 2004, with the ground's highest successful fourth-innings total being the 344-1 hit by West Indies in 1984.
Test Match Special's Geoffrey Boycott: "You'd like to think England can do this. It comes down to how they play the leg-spinner (Yasir).
"The seamers will be decent like ours, but when the shine goes it's not easy. So they'll do their bit.
"But we come to the wrist spinner - if England play as poorly as the first innings, then it's bye bye. You want England to play better, but they need Alastair Cook to hang around."
Pakistan coach Mickey Arthur on Sky Sports: "A lead of 300 would be very nice tomorrow.
"Misbah's innings in the first pleased me, Asad Shafiq has been amazing at number six, but I'm disappointed that none of our batsmen pushed on today.
"Misbah is a statesman, a very good leader of men."
England face a difficult run chase against Pakistan after the tourists ended the third day of the first Test with a lead of 281.