The 26-year-old Croatian frontman has been heavily linked with a move away from Ibrox during the transfer window, with numerous Barclays Premier League clubs interested in his services.
The Liverpool Echo reports that Fulham, Queens Park Rangers, and West Ham are all monitoring Jelavic's situation, but recent comments from the SPL top goalscorer's father suggest that he has his heart set on a switch to Anfield.
"Yes, we know about the interest of West Ham, and Queens Park Rangers," Philip Jelavic told The Scotsman. "But Nikica has five or six clubs interested, including that of the great Liverpool.
"As for West Ham and QPR, I think that Nikica is more ambitious than that. I think it’s most likely that he will sign for Liverpool. But there are still a few days until the end of the transfer window, so we will wait and see."
Reds boss Kenny Dalglsish has spent big on strengthening his forward line over the past twelve months, not least with the club-record £35million signing of Andy Carroll.
However, Carroll has failed to rediscover his best form on Merseyside, and with Luis Suarez ruled out of action until February, Dalglish may be tempted to bring in Jelavic before the end of the month.