The English newspaper suggests club owner Roman Abramovich pledged to break the bank for the centre-back.
Conte has found working with the defence at his disposal extremely difficult and has never disguised Bonucci is his dream signing.
Today Napoli President Aurelio De Laurentiis confirmed Chelsea made an offer of £58m on transfer deadline day for Kalidou Koulibaly, but it was rejected.
Abramovich now seems prepared to increase that further to £60m for Bonucci.
The Italy international is also wanted by Manchester City boss Pep Guardiola, but Juventus remain adamant he is not for sale.
Bonucci has a relatively rare skill-set for a centre-back, as he starts moves with Andrea Pirlo-esque passes and can play in three or four-man defences.
The world transfer record for a defender is £50m, the sum Paris Saint-Germain paid to get David Luiz, although the player returned to Stamford Bridge this summer for £30m.