Llori is currently playing for Portugal at the Under-20 World Cup in Turkey and the Reds are a long way down the road in terms of negotiations with both the player and club with an offer of just under £3million already on the table.
The Reds had hoped to have concluded a deal by now but talks have stalled, reportedly due to Sporting refusing to negotiate with his agent, and Liverpool are growing increasingly frustrated.
Sporting may want around £4million for the 20-year-old and boss Brendan Rodgers is likely to be able to afford that amount as he bids to add more strength in depth in defence over the summer.
Meanwhile, scouts have also been watching Deulofeu, who is in action for Spain at the U20 World Cup and they have been impressed with the 19-year-old's performances.
Barca are thought to be keen to keep hold of the player, who scored 18 goals for their B team last year, while Spurs have also been credited with an interest in the winger.