Hitzlsperger hopes to restart career at Everton
Assistant manager Steve Round told the club's website that the 30-year-old, who previously played for Aston Villa and West Ham United in England, was training with the side.
"Thomas is a long way off playing. We will assess him over the next couple of weeks but I have been very pleased with him," he said.
"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."
Hitzlsperger played 52 times for Germany but has been without a club since leaving VfL Wolfsburg at the end of last season. As well as his spells at Villa and West Ham, he also played for Lazio and most successfully at VfB Stuttgart.