Nuno stepped down on Sunday, after just over one season in charge.
Neville joined Valencia as assistant manager in July, having previously been first-team coach at Manchester United.
The former Everton captain, 38, will support interim coach Voro until a new manager is appointed, the Spanish club said on Monday.
They both took training before Valencia's Copa del Rey last-32 tie at Barakaldo on Wednesday.
Valencia's next La Liga match is at home to champions Barcelona on Saturday.
Nuno, 41, announced his departure after Sunday's 1-0 defeat at Sevilla, which left Valencia ninth in La Liga.
They are third in Group H of the Champions League, still with a chance of progressing to the knockout stage. However, the Portuguese said expectations had not been met.
"We are all in agreement that the current situation is not good, and that we must all do what is best for the club," he said. "This has always been my intention."
Valencia finished fourth in La Liga last season but have struggled this term, winning five out of 13 games.
The club's Singaporean owner, Peter Lim, has a stake in Salford City, the non-league club co-owned by Neville with his former Manchester United team-mates Ryan Giggs, Gary Neville and Paul Scholes.
Former Manchester United and England defender Phil Neville has helped take Valencia training following the resignation of Nuno Espirito Santo.