The 31-year-old is a free agent after his contract with Manchester City expired this summer.
Santa Cruz spent last season on loan at Malaga, gaining some more Champions League experience, and Hamburg believe he could aid their push for a place in Europe next season.
"Roque has been made a financially very good offer and he has told us he finds Hamburg very interesting," said the club's director of sport, Oliver Kreuzer, to the Bild newspaper.
Hamburg have now given Santa Cruz 10 days to decide whether or not to accept their offer.
"Otherwise, we have got to look at other options," added Kreuzer.
The Paraguay international netted 31 times in 155 appearances for Bayern Munich before enjoying a successful two years with Blackburn, earning a move to Manchester City.
However, he was unable to hold down a regular berth in the City side and spent more than half of his four years with the Premier League club out on loan.
The 31-year-old could still make a positive difference, though, in the eyes of Hamburg playmaker Rafael van der Vaart.
"It would be a great signing if Roque were to join us," said the Dutchman. "His experience would improve the quality of our side.
"I rate Santa Cruz very highly. He has got what it takes to fire us into Europe."