The German U-19 international is currently enjoying a loan stint at Hertha Berlin, where he has been impressive.
Brendan Rodgers has been watching his progress closely, and is thought to have made him a top priority for the end of the season.
In a bid to avoid another botched transfer window the Northern Irishman has sanctioned an early offer for the 20-year-old, which is understood to be worth around £2m.
Wolfsburg are yet to respond to the bid, but the Anfielders are hopeful that they can push through a deal.
Liverpool have stepped up their pursuit now after getting word that Hertha are plotting to snatch their current star on a permanent basis.
Cigerci is a holding midfielder who has been described as “dynamic, technically good and constantly on the move” by his current manager, and has been tipped for a big future in Germany.
The Reds endured a difficult January transfer window, in which they missed out on top targets Mohamed Salah and Yevhen Konoplyanka.