Becks, 33, wanted to train with Milan to stay fit for England during the MLS close season.
But AC wanted him on loan and all that remains now is to agree if Becks stays till the end of the season or until the start of LA’s pre-season in March.
A top Milan source said: “It was Capello’s endorsement that convinced Milan to sign Beckham.
"He told Milan vice-president Adriano Galliani ‘Beckham is a very good player and if you sign him you will see how good he still is’.
“That’s all the club needed to hear. From then on Milan focused on turning Beckham’s stay into a loan.”
England boss Capello is a legend in Milan after leading them to four Serie A titles and the European Cup in the 1990s.
Becks will play exhibitions for LA in Australia and New Zealand in December before linking up with AC.
Milan vice-president Galliani said: “He’ll stay at Milan on loan for a few months. We’re talking with his agent.”
If Becks stays for two months only, he could play for AC for free. The Milan source said: “His salary will already be paid by the MLS.”