The Yellow Submarine, who were promoted from the Segunda last season, had been knocking on the door for a Champions League position following a strong first few months.
However, their form has dipped in recent weeks, with the club now 10 points from fourth-placed Athletic Bilbao and two points behind sixth-placed Real Sociedad in the La Liga table.
"It's time to turn our situation around, to recover our intensity and the level of play we showed in the first half of the season," Inside Spanish Football quotes Asenjo as saying.
"We have to do that on the pitch, to turn around our current form."
Asenjo is currently on loan at Villarreal from Atletico Madrid.