On a quiet, leafy street of multimillion-dollar properties, one stands out: a 14,400-square-foot mansion that looks like a villa plucked from the hills of Tuscany and transplanted to Austin, Texas.

This is where Elon Musk, 53, the world’s richest man and perhaps the most important campaign backer of former President Donald J. Trump, has been trying to establish the cornerstone of an unusual family compound, according to four people familiar with his plans.

Mr. Musk has told people close to him in recent months that he envisions his children (of which there are at least 11) and two of their three mothers occupying adjoining properties. That way, his younger children could be a part of one another’s lives, and Mr. Musk could schedule time among them.

Directly behind the villa is a six-bedroom mansion that Mr. Musk helped purchase, according to two of the people and public records. The total cost of both properties was about $35 million. When in Austin, he often stays at a third mansion about a 10-minute walk away, the people said.

Three mansions, three mothers, 11 children and one secretive, multibillionaire father who obsesses about declining birthrates when he isn’t overseeing one of his six companies: It is an unconventional family situation, and one that Mr. Musk seems to want to make even bigger.

A proponent of in vitro fertilization, Mr. Musk believes strongly in increasing the world’s population. He has even offered his own sperm to friends and acquaintances, including the former independent vice-presidential candidate Nicole Shanahan, according to two people familiar with his offer. Ms. Shanahan turned him down.