The 21-year-old was a target for the Premier League giants last year, after attracting attention following a series of impressive performances while on loan at Leece.
His record of seven goals in 29 appearances prompted interest from Arsenal and United for a player labelled the 'Colombian Cristiano Ronaldo'.
But the report, quoting sources in Italy, claims Napoli have stolen a march on their rivals and have already opened talks with Udinese to sign the forward.
Muriel joined Udinese from Deportivo Cali in 2010, and after last season's loan spell, burst onto the first-team scene in Udine this term - scoring four times so far.
His good form has led Napoli to move first, and the Serie 'A' club who lie second in the table, are eager to bring him to the Stadio San Paolo.
Napoli have a strong record when it comes to signing players from Udinese, and their relationship with the club could help them again on this occasion.
The two team's have reached agreements for Pablo Armero and Switzerland international Gokhan Inler in the recent past.
Muriel signed a new five-year contract with the Italian club in September last year, and the Colombian has three caps for his national team, scoring one goal.
He's part of a hugely talented Colombian front line, including Atletico Madrid star Radamel Falcao and another of Manchester United's transfer targets, Porto playmaker James Rodriguez.