BILD says Holtby chose to return to Germany, resisting offers from Villa and Dinamo Moscow in the process.
"Lewis is a suitable strong technician who suits our game well and will add to it with his creativity," said HSV boss Dietmar Beiersdorf.
"He has already collected a weight of experience at only 23 years of age, both in the Bundesliga, the Premier League and internationally.
"That's a lot of experience that he should bring to us."
Holtby can sign permanently for HSV next June for €7 million.
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