The 25-year-old midfielder is out of contract at the end of the season and has been strongly linked with a move to England with the likes of Stoke, Sunderland and Fulham said to be monitoring his situation.
Hamburg are trying to tie the South American down to a new deal, but the wages on offer are proving an issue.
"I love Hamburger SV. But I will not accept this offer I have received," he told Hamburger Morgenpost.
"I know all about my value to the market and if we cannot agree, I will have to go."
Hamburger SV general manager Oliver Kreuzer looks unwilling to increase his offer.
"Tomas Eduardo Rincon has been handed a good contract offer. I don't have a queasy conscience. Either he is accepting it. Or he does not."
Meanwhile, Turkey international midfielder Hakan Calhanoglu has once again insisted there is no truth in stories linking him with a move to Galatasaray.
The highly-rated 19-year-old is being strongly linked with a move to the Istanbul giants.
Calhanoglu told spox.com: "There is no truth in those rumours. I am not in negotiations with Galatasaray regarding a transfer. I am a player of Hamburger SV!"