The Belgian international has been identified by Jurgen Klopp as Fabiho's successor.
Liverpool are set to make an improved offer for Southampton midfielder Romeo Lavia, which the Saints are likely to accept, according to The Athletic.
The club had an opening bid worth around £37million ($44.5m) for the Belgium international which was rejected.
Southampton, who were relegated to the Championship in May, are holding out for an overall package in the region of £50m.
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However, negotiations are ongoing and there is now an increasing likelihood of a deal being agreed. It is possible that with add-ons, Liverpool will take it to a figure that Southampton deem acceptable.
Personal terms will not be an issue, although Chelsea also retain a firm interest in Lavia.
The 19-year-old played 45 minutes for Southampton in their 3-2 friendly defeat by Bournemouth at St Mary’s on Tuesday night.
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Asked about Lavia’s future, manager Russell Martin said: “I thought Romeo, considering what’s going on, to play tonight and do all that stuff, it was good for him to get on the pitch as he has not had the minutes other people have had."
“You can ask for an update but I can’t give you one because I don’t know. I’m going to speak to (sporting director) Jason (Wilcox) now but I’ve been preparing for this game.”
Liverpool returned to Merseyside on Monday after a training camp in Germany. Jurgen Klopp’s squad fly out to Singapore on Thursday for a week-long trip which involves games against Leicester City and Bayern Munich.
Liverpool are also set to lose captain Jordan Henderson, who has agreed a move to Saudi Pro League side Al Ettifaq. They had been prepared to see Fabinho depart for Al Ittihad too, but that move is in jeopardy.
The Merseyside club, meanwhile, have also been interested in Crystal Palace’s Cheick Doucoure, Fiorentina’s Sofyan Amrabat, Bayern Munich’s Ryan Gravenberch and Manchester City’s Kalvin Phillips.
Lavia made 29 appearances as Southampton were relegated last season, missing a chunk of the campaign due to a hamstring injury. He was also subject to a £50m deadline day approach from Chelsea last September, despite only having joined Southampton a few months earlier.
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