Jurgen Klopp is keen to bolster his midfield options further despite already bringing in Naby Keita and Fabinho during the current transfer window, in addition to opening talks with Stoke City over a move for Xherdan Shaqiri.
Krychowiak spent last season on loan at West Bromwich Albion, with his arrival considered a major coup for the Baggies at the time, although he could not prevent the club from suffering relegation to the Championship.
The 28-year-old is deemed surplus to requirements at the Parc des Princes as PSG look to sell players to fall in line with Financial Fair Play regulations, and Birmingham Live reports that Liverpool are considering a move.
However, the Reds could face competition for his signature from Lokomotiv Moscow, who are also monitoring the Poland international's situation in the French capital.
Krychowiak, who played 31 times for West Brom last season, featured in all three of Poland's group games at this summer's World Cup.