The Colombian international has long been on the Reds’ radar, with rumours suggesting that their interest was strong last summer.
However after they managed to keep Suarez at Anfield, the Merseysiders’ desire to snap up the Serie A goal-getter cooled.
But it is understood that Liverpool representatives have been in attendance for Udinese’s recent fixtures, with keeping an eye on Muriel believed to have been their task.
The South American star has only managed three goals in 14 league outings this term, but is still admired by Brendan Rodgers.
Although Suarez committed his future to Liverpool before Christmas by signing a new contract, a summer move away from the club cannot be discounted.
It is said that his latest deal contains a £60m-£70m release clause which could be triggered by a foreign club.
Real Madrid are thought to be keen, and are known to have the funds to push through such a transfer.
The 27-year-old appears to be happy at Anfield once again after a turbulent 2013, but if the club miss out on Champions League qualification he may be open to an exit.
Even if Liverpool keep Suarez, Rodgers may still be open to adding Muriel to his ranks, with attacking reinforcements seen as a priority.