Express Sport understands Manchester United are courting Neymar as they aim to improve on this season's sixth-placed Premier League finish.
The Brazil international endured a difficult campaign at Barcelona, scoring 20 goals from 45 games – 11 less than last season.
Valverde will be aiming to boost Barcelona's squad this summer after the club lost out on the La Liga title to fierce rivals Real Madrid.
Reports in Spain have suggested that the former Athletic Bilbao boss could cut as many as nine first-team players from his squad as he aims to make them the division's leading powerhouses once again.
And Diario Gol claims that he has a plan if Neymar leaves the Nou Camp.
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The report says that Valverde wants to keep the Brazil international for at least one more season – but will draw up a list of rules to ensure he is fully fit and firing next season.
He will not be allowed to return to Brazil as frequently as he did under Luis Enrique, with senior players annoyed at his decision to return to Brazil while the club missed their 2-1 defeat to Deportivo La Coruna back in March.
Valverde will supposedly also ban him from going out partying, with Neymar reportedly a fan of the nightlife Catalonia has to offer.
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If he is unable or reluctant to follow those rules, and United follow up their interest, then it is claimed Barcelona will go all out to sign Dybala from Juventus.
Barcelona will reportedly move for Paulo Dybala if Neymar leaves
The Argentina international currently looks unobtainable after signing a new five-year deal with the Serie A champions last month.
But Barcelona harbour hopes of pairing him alongside compatriot Lionel Messi.
Neymar ended a rocky campaign on a high by firing Barcelona to the Copa del Rey on Saturday.
It means the 25-year-old has now won eight major trophies in four years at the Nou Camp.