The 25-year-old has scored 37 goals and registered 22 assists in 194 appearances for Los Blancos, but he has struggled to show his best form for the club since suffering a serious knee injury in July 2019.
Asensio only has two years left to run on his contract in the Spanish capital, and there are allegedly a number of Premier League clubs, including Tottenham Hotspur and Liverpool, keeping an eye on his situation.
However, according to TodoFichajes, Arsenal are in pole position for the Spain international, with the Gunners making contact with Real Madrid to register their interest.
The report claims that Carlo Ancelotti's side would be open to accepting an offer in the region of €40m (£34.4m) for Asensio, who was a late substitute against Alaves in his side's 2021-22 season opener.
The Mallorca youngster, who represented his country at the Tokyo Olympics, has won two La Liga titles and two Champions League crowns during his time at the Bernabeu.
Arsenal paid Real Madrid a fee in the region of £30m to sign Martin Odegaard at the end of last week.
Sparta Rotterdam 
Ajax 
Roma 
Parma 
Dundee Utd 
Celtic 
Valencia 
Alaves 
PSV 
Feyenoord 
Bordeaux 
Rennes 
ST Etienne 
Marseille 
Monaco 
Everton 
Chelsea 
Leicester 
Wolverhampton 
Manchester Utd 
Bournemouth 
Real Madrid 
Sevilla 
Cremonese 
Sampdoria 
Tottenham 
Liverpool 
VFL Wolfsburg 
Borussia Dortmund 
Las Palmas 
Espanyol 
Monza 
Juventus 
Lens 
Paris Saint Germain 
Fiorentina 
Udinese 
Benfica 
Estoril 
Inter 
Como 
Manchester City 
Fulham 
Nottingham Forest 
Motherwell 
ST Mirren 
Rangers 
Carrarese 
Arsenal 
Ipswich