The Merseyside club have been chasing the Egyptian star for some time, and looked set to announce his arrival during the first week of the current transfer window.
However, despite claims that a swoop was 99% complete they have been unable to push through an agreement.
Initially, it was believed that Liverpool had settled a fee in the region of £7m for Salah, but amid rumoured interest from Zenit Saint Petersburg, Basel’s valuation is now said to be north of £12m.
But, Rodgers’ side are now ready to up their offer to close to £10m, which could be the compromise figure the Swiss side settle for.
Salah – dubbed the ‘Egyptian Lionel Messi’ – is believed to be extremely keen on a switch to Anfield, which could see a deal finalised before the deadline passes at the end of next week.
The 21-year-old has been in excellent form this season for Basel, and rose to prominence in England after netting in both of his side’s Champions League group stage clashes with Chelsea.