The Brazil international is set to attract a massive January offer from Barca next month - with talk of the Catalans willing to go to €140m for the midfielder.
However, El Mundo Deportivo says Real Madrid have jumped into the battle with the January market just weeks away.
Contact with Coutinho's agents, including Kia Joorabchian, has been established and worrying for Barca is the spending power Real are now flexing.
While they haven't tabled a bid to Liverpool, Real's contract offer made to Coutinho is superior to what Barca are offering - as is the commission proposed to his agents.
Both Real and Barca are pushing to sign Coutinho next month, knowing his price could jump even higher if he enjoys an impressive World Cup over the summer.