At tea, Australia were 293 for four in their second innings - a lead of 180 - with England still to bat again.
Australia vice-captain Clarke was 73 not out and left-hander North unbeaten on 64 after they'd batted through the second of the day's three sessions in a stand so far worth 132.
England, looking to go 2-0 up in the five-match series with two to play ahead of next week's fourth Test at Headingley, had a chance for victory after reducing Australia to 172 for four - effectively 59 for four - at lunch following the exit of Michael Hussey for 64, caught behind off Stuart Broad.
But the swing that had proved so potent for them earlier in the match was missing and, in good batting conditions, their attack struggled.
That England had a shot at winning was remarkable given that two days' play had been lost to bad weather, including Saturday's complete wash-out.
But Clarke, whose footwork was a feature of his innings, stiffened Australia's resistance with several driven boundaries off Graham Onions.
However, there was a worrying moment for England when Flintoff, who has been struggling with a right knee injury, saw his left foot give way underneath him as he fell in his delivery stride.
Fortunately for England, Flintoff - the bowling hero of England's 115-run second Test win at Lord's - continued his spell.
Strauss brought on occasional medium-pacer Ravi Bopara and his second ball almost yielded a wicket when Clarke, on 38, pulled fiercely only for the captain to drop the tough chance at short mid-wicket with Australia 106 ahead.
And that was as close as England came to breaking the stand and reigniting their victory bid.
They took the new ball as soon as it was available, after 80 overs, with Australia 136 in front.
But both North and Clarke completed fifties featuring seven fours each.
And North's single off James Anderson saw the duo post a century stand.
Left-hander Hussey (18 not out overnight) fell to Stuart Broad's 12th ball of the innings - and first from around the wicket - when he edged to wicket-keeper Matthew Prior after two-and-a-half hours at the crease.
Earlier, Anderson - with his sixth ball Monday - had Shane Watson caught behind for 53. It was the makeshift opener's second fifty of the match after he'd come in for dropped left-hander Phillip Hughes.
Ashes holders Australia resumed Monday on 88 for two, still 25 runs behind.
England had all but ended any Australian hopes of victory on Sunday with Onions dismissing Simon Katich and Graeme Swann's classic off-break bowling Australia captain and star batsman Ricky Ponting for five.
England had been struggling at 168 for five on Sunday in reply to another inadequate Australia first innings, this time of 263.
But Flintoff revived the hosts with 74 off just 79 balls in a total of 376 as England built a first innings lead of 113.
Australia 1st Innings 263 (S Watson 62; J Anderson 5-80, G Onions 4-58)
England 1st Innings 376 (A Flintoff 74, A Strauss 69, S Broad 55, I Bell 53; B Hilfenhaus 4-109, P Siddle 3-89)
Australia 2nd Innings (overnight: 88-2)
|S. Watson||c Prior||b Anderson||53|
|S. Katich||c Prior||b Onions||26|
|R. Ponting||b Swann||5|
|M. Hussey||c Prior||b Broad||64|
|M. Clarke||not out||73|
|M. North||not out||64|
|Extras||(b4, lb2, w1, nb1)||8|
|Total||(4 wkts, 93 overs, 389 mins)||293|
To bat: G Manou, M Johnson, N Hauritz, P Siddle, B Hilfenhaus
Fall of wickets: 1-47, 2-52, 3-137, 4-161
Match position: Australia lead England by 180 runs
TV umpire: Richard Kettleborough (ENG)
Match referee: Jeff Crowe (NZL)
Source: AFP European Edition