The offer would have smashed Tottenham's record transfer of ?16.5m for Luka Modric in 2009 but underlines chairman Daniel Levy's determination to land a marquee signing before the transfer window closes at 11pm on Monday night.
Spur of the moment? Sergio Aguero is the subject of a record ?38.5m bid from Tottenham
Atletico's rebuttal, though, follows hot on the heels of Levy having a ?23m offer for Newcastle 's Andy Carroll rejected and gaining the same response from Athletic Bilbao's Spain international Fernando Llorente after bidding ?26m.
Levy, who has allowed Robbie Keane (below right) to join West Ham on loan, wants another striker to bolster his club's assault on Europe and the Premier League and Aguero is regarded as one of the world's best.
The 22-year-old Argentina international is the son-in-law of Diego Maradona and has been the subject of strong interest from Chelsea.
The club's majority shareholder Miguel Angel Gil claimed only last week that Chelsea had had a ?52m bid rejected for Aguero plus Atletico's Uruguayan defender Diego Godin.
Chelsea distanced themselves from talk of an offer before moving for Fernando Torres instead.
Tottenham manager Harry Redknapp watched Atletico against Real Madrid last week but initially had a stronger interest in Aguero's strike partner Diego Forlan.
However, the former Manchester United man is considered too expensive for a 31-year-old as Atletico want ?10m and his wages are ?85,000 a week.
Levy instead has opted to chase the younger Aguero as he represents better value but, although they have matched the buy-out clause, Atletico still wanted add-ons to take the deal higher.
A deal may yet be revived today but Tottenham look likely to switch to other targets.
Spurs' shortcomings at the end of the pitch were exposed on Sunday at Craven Cottage as Fulham ran out comfortable winners against a side rocked by the first-half dismissal of captain Michael Dawson.
Dawson will now be missing for the next three games and Younes Kaboul has been ruled out for six weeks after Redknapp revealed the centre-half will need an operation on his knee.
Apart from Dawson, that leaves the Spurs boss with Sebastien Bassong as his only fit first-team centre-back as long-term absentees Ledley King and Jonathan Woodgate are out and William Gallas is struggling with a hip problem.
Redknapp, who landed Rafael van der Vaart in an 11th hour deal from Real Madrid on the previous deadline day, says he does not intend to make any last-ditch attempts to sign a defender before 11pm.
'We are struggling at the moment because we are down to two central defenders,' Redknapp said. 'We have one spare defender on our books in Vedran Corluka. There is no-one else who can play in the back four.
'It's not ideal but we're not going to find somebody. Who is going to find someone good enough to play in the Premier League before Monday night?'
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Source: Daily Mail
Source: Daily Mail