The 35-year-old made his Valencia debut back in 1996 and - save for two seasons on loan at Villarreal - has spent his entire career at the Mediterranean club.
He has previously indicated he would seek an new challenge at the end of the season, while uncertainty over who will coach the team next term - Ernesto Valverde has still not signed a new contract - had seemingly done little to convince the veteran to stay.
Yet Albelda told reporters this week that his plans remain up in the air.
"This could be my last home game but it may not be," he said.
"On a personal level I have to make my the hardest decision of my career: either opt for a change of scenery, retire, or see what offers the new board make me.
"At the end of the day this club, and our aim to get into the Champions League, must come first.
"We'll see what happens when we know who the coach is going to be, but I do not want to be a burden."
Valencia are in a fierce scrap with Real Sociedad to secure the fourth and final Champions League berth, with the sides locked on 62 points with two games remaining.
Three points are the least that is required from tomorrow's home with Granada, who with a five-point cushion to the bottom three are not yet safe from the drop.
Granada defender Pape Diakhate says his team will not let their performance levels drop in their final two games, even though it would take a hugely unfortunate turn of events to see them dragged into the mire.
"We still haven't achieved survival, we have two games left, and one of them will be very difficult," said Diakhate.
"We are professionals, we're not safe yet and we have to keep working as we have been, we still need to pick up points to survive."