The 19-year-old midfielder has emerged as the top target for the Reds following the departures of Jordan Henderson and Fabinho.
Lavia has agreed personal terms with Liverpool and is keen on moving to Anfield in the ongoing transfer window.
However, negotiations between the two clubs have been tense and Southampton have already rejected two offers from the Reds.
There are suggestions that Liverpool are considering alternate targets, but according to Belgian journalist Sacha Tavolieri, the Merseyside giants are playing games with the Saints.
Lavia is the player they want but are keen to bring down Southampton’s asking price.
The Saints want a fixed fee of £40m other than the add-ons before agreeing to sell the midfielder.
Liverpool are not willing to go to that figure and are threatening to look at other options.
It has been claimed that the Premier League club are playing high-stakes poker to push Southampton to lower their demands.
