German outlet Spiegel and Spanish newspaper El Mundo quote the EIC network as claiming Real considered a €250m deal for Messi in 2013.
That would have triggered his release clause.
Inigo Juarez, then a lawyer to the Messi family, supposedly informed the player’s father Jorge of Madrid’s interest.
It is claimed Real would have been willing to pay Messi €23m a year until 2021, with €1m going to his father.
But Real have hit back in a statement today.
“This information doesn’t conform in any way to reality, it is totally false,” they said.
According to the two reports, Real made their interest known through the intermediary just weeks before they signed Gareth Bale from Tottenham.
Bale became the world’s most expensive player when he left White Hart Lane for the Bernabeu in an £85m deal.
Spiegel and El Mundo claim a meeting in a private jet between representatives from Real and Messi was scheduled and that president Florentino Perez was to be present.
El Mundo claim Real have actively pursued Messi three times during his career – the most recent of which was last year.
Lionel Messi is Barca’s talisman and club captain and he recently signed a new contract at the Nou Camp.
The club made sure his release clause was lifted to €700m (£626m) to stave off lingering interest from their rivals.
Barca were stung in the summer when PSG secured the hostile transfer of Neymar after triggering his £200m release clause.
As for Real, Bale has gone on to win three Champions League titles at the Bernabeu – but his future is in doubt.
Persistent injuries has seen the Wales international linked with a move away from the Spanish capital as Perez loses patience in their club-record signing.
Spiegel also claim Messi paid €12m to the Spanish tax authorities at he end of 2016 after a long-running investigation.