Per The Times' Matt Hughes (h/t ESPN FC), Lallana's excellent performances this year have put him on the radar of the elite European trio, who could be set to fight it out for his signature in the summer if he fails to agree a new deal at Anfield.
The England international's contract expires in 2019, and talks over a new deal have yet to begin.
Lallana arrived at Anfield in 2014 and struggled to replicate the strong form he enjoyed at Southampton on a consistent basis.
However, he has perhaps been the Reds' best player this season and has established himself among their most important attacking and creative talents, racking up seven goals and as many assists in 18 Premier League games.
The 28-year-old was especially impressive in December, as Liverpool illustrated:
Liverpool FC @LFC
Adam Lallana! 👏👏👏👏👏 Adam Lallana! 👏👏👏👏👏 @StanChart's December Player of the Month! https://t.co/t8CoAu9W6w2017-1-7 01:33:04
Along with an improving eye for goal, Lallana's tidy and creative passing and excellent technique make him an invaluable asset when the Reds have possession.
So too does his intelligent movement, as football writer Jack Lusby noted:
Jack Lusby @jacklusby_
Divock Origi was excellent in the buildup there, and Adam Lallana's third-man runs have been brilliant this season.2016-12-27 17:51:37
A versatile performer, Lallana has largely played in central midfield this season but can also play in a more advanced role or out wide.
Adam Bate @ghostgoal
Didn't win, obviously, but some effort by Adam Lallana. Topped the day’s running stats again. Twice inside 48 hours. https://t.co/AtUmimaBnH2017-1-2 20:31:56
That clubs of Barcelona, Juventus and PSG's calibre might be interested in him is a testament to how well he's played this season, though whether he could command a starting berth at any of them is another matter.
All three sides have a wealth of talent in midfield, so barring a high-profile departure at one of them, it's likely Lallana would struggle for regular minutes on the pitch.
Liverpool's ongoing transformation into title contenders under Jurgen Klopp might also see him want to stay put, though the chance of joining such prestigious clubs will no doubt prove highly tempting if any offers are made for him.