Giovanni Branchini has confirmed to Relevo that the Tuscan coach "had everything, everything, everything done" with Real Madrid.
Allegri rejected the offer "with pain in his heart".
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"He decided not to do it and it was a painful act for both him and me," said Branchini.
"He did not sign for another club that offered him more, but he chose to stay in Italy because two things coincided: a family issue and Juventus' proposal to return.
"Both things coincided and he chose not to leave Turin."