The Colombia international has been linked with a move to the Premier League side after struggling for game time at the Bernabeu this season.
But Marca reports that Rodriguez would prefer to stay put and try to win back a place in the Real Madrid team.
Rodriguez returned to the Spanish capital in the summer after a two-year loan spell at Bayern Munich, but he’s found it difficult to break into Zinedine Zidane’s plans.
The 28-year-old has struggled with injury problems as well, making just nine appearances in all competitions so far this season and scoring one goal.
Arsenal are willing to hand him a lifeline by offering him regular game time in England, but Rodriguez is understood to be determined to prove his worth in Madrid instead.
Manchester United and Chelsea have also shown interest in the playmaker in the past.
The midfielder is currently sidelined with an injury problem he sustained during the November international break.
Real Madrid travel to Alaves on Saturday after being held to a 2-2 Champions League draw by PSG in midweek.
Zidane’s side are currently second in the La Liga table, but only trail rivals Barcelona on goal difference, with both sides on 28 points.
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