The 26-year-old has been in outstanding form for Samp after joining from Serie A rivals Udinese for 10million euros, and reports claim they will be looking for at least double that fee should they agree to let him move on.
Inter Milan have also been heavily linked, but reports in Italy claim that Sampdoria will not sell to a Serie A rival, and that could put Liverpool in the box seat to do a deal when the transfer window opens.
Reds boss Jurgen Klopp has put at least one new striker on his wanted list as doubts surround the future of Daniel Sturridge, and Muriel appears to be a target.
However there is interest elsewhere, and La Liga side Sevilla could hand him the opportunity to head to Spain, while Chelsea have also been linked in the past.
Germany coach Joachim Low has claimed that Manuel Neuer's recovery is on schedule and that he is in line to make Germany's World Cup squad.