The former Brazil worldwide has spent over a decade at Anfield but is expected to seal an emotional exit amid an increased level of competition in his role.
Lazio are confident of thrashing out personal terms with the midfielder's representatives in the coming hours in order to get the deal over the line soon.
Khedira played a staring role in Juventus' run to the UEFA Champions League final last season.
Liverpool stalwart to seal £5m exit agreement reached with Serie A giants Posted by Ron Carne
A fee of £5M represents solid money for the 30-year-old, whilst shifting him off the wage bill would certainly help Liverpool balance the books, with Lucas reportedly on around £80,000 per-week.
With one year left on Lucas's deal, Liverpool couldn't expect more than £5m - so the decision now rests with the player himself.
The Brazilian has frequently been used as a defender by current Anfield boss Jurgen Klopp but was deployed as a defensive midfielder for much of his time under former boss Brendan Rodgers.
According to The Guardian weeks of negotiations between Liverpool and Hull have now completed, and the player could be announced this weekend.More news: Boyd Holbrook's Exit from 'Narcos' Confirmed, Season 3 Teaser Trailer Released