The 30-year-old, who has also played for Aston Villa and West Ham United, is currently a free agent after departing Wolfsburg in the summer.
He joined the Toffees on trial last week, but Round has insisted that the club are yet to decide whether to offer the German a contract.
"I have been very pleased with him. Thomas is a long way off playing but we have given him the opportunity to come in and it is up to him now. We will assess him over the next couple of weeks," Round told the club's official website.
Hitzlsperger made 52 appearances for his country.