New Reds boss Roy Hodgson is thought to be an admirer of the versatile 30-year-old, who left Wigan after his contract expired at the end of last season.
Hodgson tried to sign the Austrian on three occasions when manager of Fulham, but failed in each attempt.
It was also thought that the 62-year-old was trying to secure his signature at Craven Cottage before moving to Anfield at the start of July.
Aston Villa, Sunderland, Bolton and Blackburn are also rumoured to be interested in landing Scharner's signature, but Liverpool is believed to be his preferred destination.
If the deal is to be successful, Scharner could possibly fill the void left by defensive midfielder Javier Mascherano should the Argentine depart.
"Roy Hodgson and Paul Scharner - this is a special relationship," said Scharner's agent, Valentin Hobel.
"Hodgson thinks that Paul's strengths in midfield are better than his attributes in central defence.
"Liverpool would be a dream, of course, but we are in contact with several clubs. Because he is a free transfer, he can choose."