The former West Ham United man is currently a free agent after leaving Bundesliga outfit Wolfsburg last summer.
Now, he is on trial at the Merseyside club and Moyes is pondering on whether to offer him a contract with the Toffees.
The Scotsman finds himself short of midfielders at the moment due to the injury of Darren Gibson and he of course lost Jack Rodwell to Manchester City in the summer transfer window.
However, Moyes insists he will continue to assess Hitzlsperger and it could still be a few weeks before a decision is made.
The 30-year-old will still feel as though he has plenty to offer in the English Premier League and he could be a fantastic bit of business from Everton.
Everyone will remember his powerful left-footed shot that would worry opposing goal-keepers as it can be unstoppable.
It is likely that Moyes will really put Hitzlsperger through his paces and see if is up to the high standards that he sets.
Could he head back to the Premier League?
We shall see.