Brendan Rodgers is believed to be a big fan of the Argentine winger, although Tottenham are currently in pole position to land the 21-year-old.
Lamela's agent Pablo Sebbag and Roma's sporting director Walter Sabatini reportedly held talks with Tottenham's technical director Franco Baldini in London this week, but the report claims negotiations broke down over a fee.
Roma have set a £30m asking price for their second highest league goalscorer last season, whereas Spurs are reluctant to go above £25.7m. Andre Villas-Boas has also prioritised a move for Brazilian starlet Willian, leaving the door open for Liverpool to steal in.
Arsenal are also thought to be tracking Lamela, as Gunners boss Arsene Wenger comes under greater pressure to make a high profile signing this summer.
The former-River Plate starlet has been excellent since swapping Argentina for Serie A in 2011, and he's been heavily linked with a Premier League switch for the past few months.
Liverpool have been busy already this summer, signing Iago Aspas and Luis Alberto from Celta Vigo and Sevilla respectively, and Kolo Toure from Manchester City.
But, like Spurs, they've been linked with Anzhi playmaker Willian - suggesting funds remain in place to secure a big-money capture before the transfer window shuts.
The most recent reports claim Tottenham head the pursuit for Willian, which could allow Liverpool to edge ahead and seal an agreement for Lamela.