German club are reportedly considering an attempt to sign the Reds' midfielder, who would likely be forced to take a significant pay cut if he were to switch to the Bundesliga
The 29-year-old left Chelsea and joined the Reds on a Bosman transfer last summer, with a salary believed to be in the region of €100,000-a-week.
However, the ex-England man failed to produce the kind of form he had shown during his time at Stamford Bridge, and with Liverpool having brought in three new midfielders this summer, Cole is believed to be on his way out.
Now The People reports that the Bundesliga outfit are ready to table a transfer fee though if he were to move to Hamburg, Cole would have to take a significant pay cut as his salary, believed to be in excess of €5 million, is well above HSV's limit, which is in the region of €3m.
The German club made an exception for Ruud Van Nistelrooy, who earned €6m in 18 months, but are unlikely to do the same for Cole.
Another cause for question is the player's role at Hamburg, who have structured their transfer campaign around youth since the arrival of sporting director Frank Arnesen.
Cole made just nine Premier League starts for Liverpool last year, scoring two goals.