The 27-year-old has barely featured for the Reds after suffering a back injury at the start of last season.
However, his injury meant he missed 44 games for the Reds whilst Alexander-Arnold and Joe Gomez took turns to impress in his absence.
Clyne has seen his role limited at the start of this season and could well be on the way in January, according to The Mirror.
The Foxes signed attacking right-back Ricardo Pereira from Porto this summer, however he has predominantly been used as a right winger by manager Claude Puel.
Daniel Amartey has taken up the right-back spot, but it appears that Puel wants to sign Clyne to bolster his options.
Clyne joined Liverpool from Southampton in 2015, enjoying two successful seasons at the club that saw him being selected as England's starting right-back.
However, injury has put a halt on his career, restricting him to just five appearances and two starts last season. Meanwhile competition for his place has led to question marks surrounding his future.
He has featured once for Jurgen Klopp's side this season; in the Reds' 2-1 League Cup defeat to Chelsea at Anfield over a fortnight ago.