The 30-year old German is available on a free transfer after departing Wolfsburg earlier this summer and is training with the Toffees in a bid to win a contract.
Everton assistant manager Steve Rounds says that Hitzlsperger, who has had previous Premier League spells with Aston Villa and West Ham United, has been training well but it is too early to tell if he will be offered a Goodison Park deal.
"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," Rounds told the club's official website.
"He seems a really good lad and I don't think I have seen him without a smile on his face.
"What will determine whether he gets a contract is his fitness. He has had a bad injury and we would want to see that he could sustain training.
"If he could do that, he would add quality and strength to our squad. He has got a history of playing at the highest level."