The Toffees' 1-1 draw at the Britannia Stadium on Saturday left them lying in fifth place, with their opponents trailing them by only three points in ninth.
And with the margins currently so fine in the top half of the table, Moyes believes Stoke have a chance of securing an unexpected place in Europe's premier competition.
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The Everton boss is concerned that the size of his squad could be the factor that sees them fall short in their own quest to secure a top-four spot.
Moyes was particularly worried that a lack of attacking firepower could prove costly as the campaign wears on.
"We know we have not got the strongest squad, we are certainly short," he continued.
"We haven't got enough in the forward areas at the moment. I don't know if we'll actually have enough but I am not going to talk us down."