The game at the club’s famous Mestalla Stadium was marked by protests from some sections of the stands.
Murthy and owner Peter Lim have come under intense criticism this season, particularly after making the decision in September to sack manager Marcelino and replace him with current boss Albert Celades.
A statement published on Valencia’s official website read: “The Valencia CF legal department have provided the police with evidence of serious threats - including death threats - sent to the club president's mobile phone in the past hours, in light of the fact that such offences carry possible penal consequences.”
A group of several hundred Valencia supporters gathered outside the stadium to protest before the Alaves game, with signs bearing slogans such as ‘Us, passion, you commission’, and ‘When serving Valencia, don’t serve yourself’.
Murthy was reported to have gestured towards fans with his finger to his lips as he responded to chanting during the game.
There was also support in the stands for legendary former goalkeeper Santiago Canizares, who has been outspoken in his own criticism of Lim’s ownership.
After years of instability, former manager Marcelino had been a popular figure at the club after guiding Valencia to two fourth-place finishes in his two full seasons in charge.
Marcelino later said he felt he had been sacked for winning the Copa del Rey, a competition not valued by the club’s hierarchy.
Under Celades, Valencia have won two and drawn two in La Liga, also winning one and losing one in the Champions League group stage.
According to Onda Cero, Lim has scheduled a meeting with all the club's staff for next Tuesday, the purpose of which is unknown.
Valencia’s next fixture, which follows the upcoming international break, sees them travel to the capital to face Diego Simeone’s Atletico Madrid.