Valencia finished a disappointing ninth in La Liga last season, although they did reach the final of the Copa del Rey, which they lost on penalties to Real Betis.
The club have also been cutting costs in recent seasons and have sold a number of key players without replacing them.
Bordalas, who had one year left on his contract, criticised the board’s transfer strategy in January.
“Valencia CF announces that Jose Bordalas is ending his time as coach of the first team,” a statement read.
“The Club wants to thank him for all his work and dedication during his time as a Valencianista, in which the team reached the Copa del Rey final, and wishes him the best of luck in his career.”
Gattuso has reportedly been in talks with Valencia for several weeks and is set to sign a two-year contract to take over at Mestalla.
The former Italy midfielder was in charge of Napoli when they won the Coppa Italia in 2020 but was sacked the following year after missing out on Champions League qualification.
Gattuso also managed AC Milan between 2017 and 2019 but only lasted one full season as the club finished fifth.
He was due to be appointed coach at Tottenham last year but the deal collapsed after a backlash from fans around comments Gattuso previously made about women in football and same-sex marriage.
Valencia fans protested last season about the running of the club under owner Peter Lim.
Anil Murthy, the club’s president, was sacked on Monday after leaked audios recorded him criticising Lim and threatening to destroy the reputation of midfielder Carlos Soler, if Soler refused to sign a new contract.
Valencia — whose sixth and last league title came in 2004 — have finished ninth, 13th and ninth in the last three seasons in La Liga.