Manchester United have been linked with a sensational summer move, suggesting they could meet his £173million buy-out clause.
However, Jose Mourinho has since played down talk of a pursuit, insisting it is unrealistic to lure the Brazil superstar away from the Nou Camp.
But Madrid appear to have more faith a transfer out of Barcelona could happen.
Real Madrid believe Neymar could quit Barcelona
Bernabeu president Florentino Perez feels Neymar has an escape plan
According to Spanish outlet Diario Gol, Bernabeu president Florentino Perez feels Neymar has an escape plan.
The 25-year-old almost single-handedly kept Barca in the Champions League when they recovered to beat Paris Saint-Germain last month.
Although La Liga progress is stalling, especially after Neymar's red card and Barca's defeat to Malaga this weekend.
Off the pitch, Luis Enrique is leaving in the summer and no replacement has been set yet.
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Neymar is up alongside Lionel Messi now as Barcelona's star player
If the sport management of Barca is still seen as disastrous, it is said Madrid feel Neymar will jump ship.
One further reason behind this is his tense relationship with Lionel Messi.
Despite their public claims of being friends, their fight to capture the leading role at the club makes things difficult.
Madrid star Cristiano Ronaldo reportedly believes Neymar has now overtaken Messi and that the Argentina international will not stand for his attacking partner to surpass him.
Jose Mourinho's Manchester United would love to sign Neymar from Barcelona
An alternative for Neymar is United – one of few teams who would be able to afford him.
Diario Gol add the Red Devils have not stopped in their push to sign the Barca winger.
Neymar's future is likely to rely on what success the Catalan giants earn domestically and in Europe.
And even the 2018 World Cup in Russia could prove the final turning point with Neymar's Brazil now ranked No1 while Messi's Argentina are struggling.