According to a report by American media company Insider, Mr and Ms Musk filed a petition to change the name of their child in Texas.
The reason for the name change request was for the children to have their father's and mother's last names in their middle name.
Ms Zilis is the top executive and director at Neuralink, a company founded by Mr Musk.
The company was founded by Mr Musk and his business partner.
The news of Mr Musk's twins comes just weeks after it was revealed his daughter had taken steps to cut ties with him, declaring her wish to no longer be related to him in any way, shape or form.
Musk is a well-known businessman.
His personal life is not easy to understand. He had at least seven children.
His private life hasn't always stayed out of the public eye, having dated and fathered two children with a pop star.
He keeps his messy private life separate from his business interests.
It's possible that this story could be damaging because he worked with the man. He was in a position of power.
There are rules that prevent executives from having consensual relationships with employees.
McDonald's had a relationship with an employee that led to the ousting of its CEO.
The story won't help Musk's reputation with many employees at the company he is buying.
Many employees are worried that Mr Musk's erratic business practices could cause the company to fall apart.