Pizarro joined the Blues last summer on a free transfer from Bayern Munich, but has struggled to make an impact.
Reports have linked the Peruvian striker with a return to the Bundesliga at the end of the season and a move back to Bremen.
The 29-year-old spent two years with Bremen, scoring 29 goals in 56 league games, before joining Bayern in 2001.
Bremen's sporting director Klaus Allofs has confirmed he wants to sign Pizarro, but is concerned about the cost of any deal.
"I consider Claudio an interesting player," Allofs told Bild. "A player who will soon be 30 years is not too old for us.
"From a financial point of view it doesn't seem feasible, but I'm going to ask."