Genk turned down HSV's opening two proposals as insufficient, but the mood around a deal has shifted with the latest attempt.
"This final offer could be enough to reach an agreement," Tavolieri said.
The 24-year-old is contracted to Genk until 2028, leaving the Belgian club under no obligation to sell, but reports suggest the Morocco international — who featured at this summer's World Cup — is keen to make the move to Hamburg himself, a factor that could ease the negotiations regardless of Genk's strong contractual position.
El Ouahdi joined Genk from RWD Molenbeek in September 2023 for a fee of around €2.5m and has since developed into one of the club's most important players, with his market value now estimated at roughly €17m following a series of strong seasons on the right flank and growing recognition with the Moroccan national team.
He contributed 12 goal involvements in all competitions last season alone, form that has drawn interest from well beyond Hamburg — Benfica, Bayer Leverkusen, Milan and Roma have all been credited with monitoring the player at various points this window, underlining the scale of the asset Genk are negotiating over.
Should the third bid fail to land him, HSV have alternatives lined up. Arthur of Bayer Leverkusen remains an option, as does Werder Bremen's former full-back Yukinari Sugawara, with both representing considerably cheaper solutions to the same problem on the right side of defence than a deal for Genk's prized asset would require.