A bitter end to the season which has featured capitulations in the Champions League and Copa del Rey have made Valverde an unpopular figure at Camp Nou.
The former Atheltic Bilbao coach led the Catalans to La Liga glory but his future looks in serious doubt despite extending his contract to the end of next season earlier this campaign.
Club president Josep Maria Bartomeu has defended Valverde in the wake of Barcelona’s 2-1 Copa del Rey defeat on Saturday, exonerating him from the blame and reiterating his belief in the 55-year-old.
But Spanish paper Mundo Deportivo claim that Barcelona have identified four men to succeed Valverde, from club legends and national managers to out-of-contract coaches.
The Italian has departed Juventus this summer after five successful seasons in charge, claiming the Serie A title every year.
Allegri did admit he is open to taking a year out of the game, however, before returning for a different project.
Barcelona would prove exactly that, with a bigger emphasis on attacking football than what was demanded in Turin.
Emery has a strong reputation in Spain, where he guided Sevilla to three Europa League titles and had impressive spells at Valencia and Almeria.
The Spaniard is understood to be admired by Barcelona chiefs for his gradual development as a coach, which landed him the Arsenal job last summer.
The high-pressing coach has two years left on that contract in north London, and is expected to see out his deal.
The Spaniard is currently in charge of the talented Belgium national team, who finished third in last year’s World Cup.
His contract with the Red Devils runs until after Euro 2020, however, which he may want to see out.
The former Wigan and Everton manager has risen through the ranks in his coaching career and was linked with the Real Madrid job when Julen Lopetegui was sacked in October.
Koeman is a club legend at Barcelona, for whom he scored the winning goal in the first European Cup triumph in 1992.
The Dutchman is currently coaching his national team and leads them into the Uefa Nations League finals next week, having had spells in England to varying success with Southampton and Everton.
Koeman was coached by Johan Cruyff and embodies the philosophies of the late manager, which is a key requirement for demanding Camp Nou fans.