Manchester United are watching Allan's progress at Napoli this season.
The Brazilian midfielder is impressing in new coach Carlo Ancelotti's system this term.
Calciomercato.com says United have posted scouts to Italy to follow Allan's campaign.
The midfielder is regarded highly by United manager Jose Mourinho, who could be forced to find a replacement for Paul Pogba in 2019.
Allan, 27, has been with Napoli since 2015, having signed from Udinese.