The 31-year-old was targeted by the club last year, but the former Spanish international shot-stopper elected to remain in Spain.
It would now cost around £2.5million to snatch him from Deportivo, who were relegated from La Liga.
Neil Lennon must decide if he is prepared to commit this amount of his transfer kitty to filling the goalkeeping position.
He could help fund this by moving on Lukasz Zaluska and Dominic Cervi, who failed to oust loan keeper Fraser Forster last season.
But there are other names in the frame for the No.1 spot, including Forster, who is still waiting to find out if Newcastle will allow him to move on.
Shay Given remains on the list of possibles drawn up by the Celtic management and scouting team too.
But with Manchester City demanding any move for the Republic of Ireland keeper is a full transfer, and the cash-rich club holding out for a substantial fee, a return to Parkhead for Given may have to wait beyond this window.
Celtic appear to be on course for more immediate joy with another signing target, Victor Wanyama.
Germinal Beerschot's midfielder is another player Lennon has been tracking for more than a year, and a deal to finally bring him from his Belgian club now appears to be imminent.
The teenage Kenyan's agent, Ivan Modia, confirmed: "There are several clubs interested, but Celtic have not put in an offer."
Standard Liege believed they were in pole position to win the race to sign Wanyama, who turns 20 this week and who was one of the few successes in the Beerschot side which battled all last season against relegation.
A £900,000 fee had been agreed between the clubs, but Liege have now fallen behind Celtic in the contest to land the youngster who can also play in central defence.
Wanyama thought he was on his way to Parkhead 12 months ago, and was disappointed when the deal fell through at that time.
Modia confirmed Wanyama's desire to join up with Lennon remains strong and said: "The player wants to go to Celtic."
Lennon is also keeping tabs on Sturm Graz defender Gordon Schildenfeld and remains hopeful a deal can be done for Maccabi Petah Tivka striker Omer Damari.