The former Aston Villa and West Ham player has been without a club since June after being released by Wolfsburg and Toffees manager David Moyes has been monitoring the 30-year-old to see if the player can add depth to his squad.
The German has a fortnight to convince Moyes that he has something to offer him but Hitzlsperger is not fit enough to be considered for Tuesday night’s League Cup game against Leeds.
Round believes that the midfielder would add quality to the side and that the player’s fitness will determine whether he gets a contract.
“I have been very pleased with him,” he told reporters.
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“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.”
Hitzlsperger is a former Germany international and has been capped 52 times.