Southampton seemingly won't budge from their £50-million valuation for the holding midfielder - a figure Liverpool are reluctant to pay, given the 19-year-old's lack of experience at first-team level.
Last season, which was his first at the top level, saw the Belgian make only 29 Premier League appearances.
That said, he has already received his first cap for the national team, so his ceiling is clearly high.
Comparisons with Bajcetic
Interestingly, Joyce also explained that Liverpool's reluctance to pay £50 million for Lavia "is not about a lack of funds". Rather, he says, it's "about value for a player who is a year older than Stefan Bajectic".
Bajcetic, who impressed during his breakthrough campaign, has played only 18 fewer Premier League games than Lavia.
The young Spaniard is currently the only dedicated holding midfielder in Jürgen Klopp's squad. However, he fights a battle to be fit in time for the league opener at Chelsea this Sunday, and his place will likely go to one of either Alexis Mac Allister or Curtis Jones.