Reds boss Jurgen Klopp wants to sign the Belgian midfielder and Liverpool have been holding talks with Southampton to agree a deal.
They have not yet managed to find common ground with the Saints and Liverpool are a team in a hurry.
The Reds are working overtime on a deal and the start of this week will be intense in the negotiations, according to Belgian journalist Sacha Tavolieri.
Liverpool are very much keeping in mind that the new Premier League season is due to start next weekend.
The club want concrete answers on a possible deal for Lavia before their opening game, on Sunday, at Chelsea.
Lavia is keen to make the move to Anfield, but Liverpool want to know if a deal can happen on terms they find acceptable.
Liverpool are expected to have a busy period before the transfer window closes regardless of whether they sign Lavia or not, with further fresh faces tipped to come in.
