Man United deal for Real Madrid star hijacked by Man City: £68m offer matched
Rodriguez, 25, has been heavily linked with a move away from Real Madrid after a disappointing season under Zinedine Zidane.
Manchester United boss Jose Mourinho is reportedly interested in bringing the Colombia international to Old Trafford as he looks to bolster his ranks in the summer.
Spanish outlet Diario Madrista claim the Portuguese boss has stepped up his pursuit of Rodriguez after submitting a £68m offer to Real Madrid chiefs.
But it is claimed Manchester City have weighed in with a matching bid of their own.
Rodriguez is apparently open to joining either club but would prefer a move to Old Trafford.
However, the prospect of United being without Champions League football next season could be a problem with Rodriguez seeking these assurances.
United are unlikely to finish in the top four but do have the chance to qualify for the competition if they win the Europa League.
The Red Devils reached the final last night after securing a 2-1 aggregate win over Celta Vigo.
Rodriguez has made 31 appearances for Real Madrid this season and has scored 11 goals.