The German midfielder has been training with the Toffees squad for two weeks after being handed a trial with the club in September.
Assistant manager Steve Round has said that the club are keen to test the former West Ham player's fitness before making any decisions on whether he will sign a permanent deal.
"He seems a really good lad and I don't think I have seen him without a smile on his face," Round told The Daily Mail.
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"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, who was released by German side Wolfsburg in June, amassed 52 caps for his country during his international career.
He is joined in the Under-21s side by Sylvain Distin and Jan Mucha, both of whom featured in the senior side's draw at Wigan on Saturday.
Magaye Gueye and Apostolos Vellios will also feature for David Moyes' youth side.