Reports stemming from Goal suggest that Arsenal are ready to sell Olivier Giroud in order to fund a £35million move for the France international, although the Guardian piece suggests it would cost in the region of £40million to lure Benzema away from Madrid.
Manchester United goalkeeper David de Gea could also be used as a makeweight in any potential deal.
Benzema signed a five-year contact at Real Madrid last summer but the sacking of Carlo Ancelotti, as covered by the Guardian, could result in unrest in the dressing room. Not even a plea from Cristiano Ronaldo could save the Italian manager.
Rafael Benitez, who is leaving Serie A side Napoli this summer, is the early frontrunner to replace Ancelotti at the Santiago Bernabeu. If the latest appointment causes a stir in the dressing room, Louis van Gaal might find it easier to prise stars away from Real Madrid.
At Lyon, Benzema won four Ligue 1 titles. Following his move to Real Madrid in 2009, he has won the Champions League, La Liga, the Copa Del Rey, the Spanish Super Cup, the UEFA Super Cup and the FIFA Club World Cup.
Having scored 22 goals in 46 appearances this term, and 214 across a productive career, the 27-year-old is renowned as one of the hottest goal-scorers in the world and is now considered to be a target of Manchester United.
Javier Hernandez has, as reported by the BBC, been snubbed by Real Madrid and his future at Old Trafford remains uncertain. That leaves van Gaal with limited options in the final third for next season, with Wayne Rooney his only reliable source of goals.