Hitzlsperger has been without a club since leaving Wolfsburg at the end of last season but has Premier League experience following spells with Villa and then the Hammers last season.
The 30-year-old has made a positive first impression at Everton, but Blues assistant Steve Round says a contract is still some way off.
Round told the club's official website: "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.
"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."
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