The Daily Mail says Rodgers has been unsure about Pepe Reina, 30, since he arrived in the summer and wants a long-term replacement.
He almost moved to sign England international Butland in the summer but held off. Now he is ready to step up his interest in the 19-year-old when the transfer window opens.
Zieler, meanwhile, is one of the best keepers in Germany and is contracted to Hannover with a £12.2m buy-out clause.
Having started his career at Manchester United, the 23-year-old joined Hannover in 2010 and has played more than 70 games. He also has two full Germany caps.