The Express claim that Jurgen Klopp has long been an admirer of Asensio and Madrid believe including the 23-year-old can tempt the Reds to depart with their prolific winger.
Meanwhile, Le10Sport claims that Liverpool are plotting a move to bring Philippe Coutinho back to the club.
The Brazilian has endured a turbulent time since joining Barcelona 18 months ago with the player himself admitting his future is uncertain.
Rafael Camacho has joined Sporting Lisbon in a permanent deal for £5m with the deal including a buy-back clause and a sell-on percentage.
Liverpool fended off late competition from Bayern Munich to make their first signing of the window in Dutch starlet Sepp van den Berg for £1.3m.
The 17-year-old has signed a long-term deal at Anfield, with Ajax also thought to have made a late run at the talented defender.
Speaking on his move from PEC Zwolle, van den Berg said: “It’s just an amazing feeling. It is, for me, the biggest club in the world and it’s a dream come true.”
Elsewhere, Nicolas Pepe is reportedly still attracting attention, with the Daily Record’s Duncan Castles claiming that there is still “hope” for a deal to bring him to Liverpool. It had been reported that Inter Milan had lodged a £72m bid for the winger but Castles has quashed those rumours.
Marko Grujic has rejoined Hertha Berlin on a season-long loan after a successful spell with the Bundesliga side. Liverpool will receive £2m for the loan plus up to £500,000 in bonuses.
Ryan Kent’s hopes of rejoining Rangers on loan next season have been dealt a blow, with Liverpool reportedly only interested in sanctioning a permanent deal for the 22-year-old.
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