News of the World says both clubs along with Real Madrid, Barcelona, Inter and AC Milan could fight it out for Benzema in January.
The 20-year-old current contract expires in 2013 and holds an £80million releasue clause.
Lyon president Jean-Michel Aulas said: “Our aim is to win the Champions League with Karim.”
It is believed the massive releasue clause may fend of the interest from Manchester United and other clubs, but Manchester City’s new owners would still be keen on a deal.
