Bordalas was appointed as Valencia manager before the start of the 2021-2022 campaign, just two days after departing Getafe, where he'd enjoyed a successful five-season spell.
The 58year-old was chosen by Valencia owner Peter Lim as the successor to Javi Gracia, who became the 11th manager to leave the LaLiga side in the past nine years.
Bordalas has guided Valencia to 10th in the league and suffered penalty shoot-out heartbreak in the Copa del Rey final against Real Betis in April.
Valencia finished 13th in the league last season, but supporters have voiced their disgruntlement at the club, directing questions towards to Lim, who has sold top players to reduce high debts and made no signings to strengthen the squad.
Bordalas revealed his frustrations before a clash with Athletic Bilbao on Saturday, suggesting that Valencia must bring in top-level talent as opposed to selling their best players.
Los Che supporters responded by barracking their team and Bordalas, singing chants against the manager, and Valencia have urged for the fans to back their coach and squad.
"Valencia CF condemn the campaign of harassment and disruption that both coach Jose Bordalas and the team have been receiving throughout this season," a statement from the club said on Monday.
"This trend, fed by some media outlets and which has the sole purpose of harming the coach and the team, has led to very serious insults from various opposing sets of fans towards our coach.
"In addition, it also intends to create a negative environment in an attempt to try to affect refereeing decisions.
"Valencia CF request the utmost respect for both our coach and our squad."