The USA international has impressed this season under new Juve coach Thiago Motta.
Weah Snr recalled to La Repubblica: "I had spoken about Timo with (former Milan director Paolo) Maldini, there was the possibility of him going to Milan, but it didn't materialise.
"When my son mentioned Juve to me, I told him not to hesitate for a moment. We discussed it with his mother and the agent and we all agreed: we couldn't say no to an opportunity that millions of kids dream of but that only a few have.
"Regardless of my support, it's the right team for him."
Juve and Milan played out a 0-0 draw on Saturday at San Siro stadium.