The 23-year-old enjoyed a successful loan spell at the Stadium of Light during the second half of last season, and was reportedly high on Gus Poyet's list of targets in the summer transfer window.
The Black Cats are believed to have made several offers for the left-back over the past month, as well as a cash-plus-player deal including Modibo Diakité, but it now seems the Spaniard is not for sale.
With Sunderland's current senior squad not containing any out-and-out left-backs, the position is a huge problem area for Poyet at present, and is in major need of bolstering before the curtain lifts on the new Premier League season.
With a deal for Alonso now seeming unlikely, here are three possible alternatives the Uruguayan may look to to fill the vacant left-back position at Sunderland;
Like Alonso, the 24-year-old also enjoyed a successful season while on loan at the Stadium of Light.
During the 2012-13 campaign Rose made 27 appearances for the Black Cats, scoring one goal and making a noticeable impact on Wearside.
Sunderland fought to sign the Tottenham defender on a permanent basis, but to no avail with Rose being considered the perfect candidate by Spurs to fill the boots of the QPR-bound Benoit Assou-Ekotto.
But now it seems the 24-year-old could be allowed to leave White Hart Lane, with incoming manager Maurico Pochettino seemingly willing to listen to offers for the defender.
Sunderland have been mentioned among a host of possible suitors for Rose, and he may well make a return to the Stadium of Light this summer.
Reports from from the German news website Bild earlier this month claimed Sunderland were eyeing up a potential move for the defender, although the Austrian was also said to be attracting attention from both Bordeaux and Palermo.
The 28-year-old joined Schalke from Bochum in 2011 and has since gone on to make 74 appearances for the German side.
Renowned for his crossing and set-piece ability, Fuchs could prove to be a good addition to Poyet's side.
Reports suggest the Austrian is available for as little as £2million, which will no doubt only tempt the Uruguayan manager further.
A relative unknown, Sunderland are reported to have been tracking the Bosnian defender whose contract at Wisla Krakow expired this summer.
Standing at 6"5, Bunoza offers a huge physical presence and could add some much needed metal to the Sunderland defence.
Black Cats fans will hope that should the Alonso deal be dead, Poyet reacts by signing a more household name, and for that reason the 26-year-old surely may only be considered a last resort.
Who would you like to see playing at left-back for Sunderland next season?