Hitzlsperger has started a two-week trial at Everton in the hope of securing a permanent deal.
"I have been very pleased with him," said Everton assistant manager Steve Round.
"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.
"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."