Valverde was confirmed as the new Barcelona manager on Monday, with the 53-year-old now charged with the task of toppling Real Madrid as La Liga champions.
Real won the prize for the first time in five years earlier this month, beating Malaga 2-0 on the final day courtesy of goals from Cristiano Ronaldo and Karim Benzema.
Zinedine Zidane's side can top off an already successful season by becoming the first club to retain the Champions League when they face Juventus in the final on Saturday.
With Real firing on all fronts, it already looks as if Valverde faces a fight to bring the glory days back to the Nou Camp.
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Real Madrid think Lionel Messi could grow unhappy at Barcelona
Real Madrid players think Ernesto Valverde will fail at Barcelona
And Diario Gol claim their president, Florentino Perez, thinks a frustrated Lionel Messi could make things difficult for the new Barcelona boss.
Reports have claimed that the Argentina international is already unhappy, with team-mate and close friend Jordi Alba one of the players Valverde could show the exit door.
Messi is said to be a key voice at the club and Barcelona chiefs are doing everything to keep him happy, with his current contract expiring next summer.
And Real players are said to have compared his appointment to that of Rafa Benitez.
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Ernesto Valverde has been unveiled as the new manager of Barcelona
In a shock move, the Spanish giants sacked Carlo Ancelotti and replaced him with the current Newcastle boss – just one year after Ancelotti had steered them to the Champions League trophy.
Benitez struggled to win over senior players during his short stint at the Bernabeu, with Cristiano Ronaldo and Sergio Ramos reportedly at odds with his appointment.
As a result, they think that Valverde faces a similar fight at Barcelona, which could lead to Real continuing their dominance of both Spanish and European football.