Dos Santos was hailed as one of the brightest prospects in world football when he signed from Barcelona four years ago, but struggled to make an impact and has had subsequent loan spells at Ipswich, Galatasaray and Racing Santander.
The forward, who has made just 15 starts during his time at White Hart Lane, scored and helped set up two more goals in Spurs' FA Cup win over Cheltenham earlier this month, but is not in Harry Redknapp's long-term plans and the player himself wants to leave.
Villarreal, who sit just above the relegation zone in La Liga, appear to be his preferred destination.
The club's vice-president told elperiodico mediterraneo: "The negotiations are going well and it seems the player wants to come to Villarreal."
The Yellow Submarine face competition from La Liga rivals Granada for Dos Santos' signature, however.
"We are the only Spanish club that has made a formal offer for him but we are running out of days before the end of the transfer window and it could be that another club will come in tomorrow [Thursday] with more money," Granada president Quique Pina told Granada Hoy.
"There are players out there of a high level and we found difficulty bringing them to Granada.
"We are fighting as hard as we can and at this moment and we are in a good position over Giovani so long as another club does not get involved.
"It's logical that the player wants to play for more important clubs but if there isn't a better offer it will be difficult for him to go to another club."
Dos Santos is just one of a number of fringe players who may leave Spurs before the transfer window shuts next Tuesday.
Heurelho Gomes, Niko Kranjcar, Steven Pienaar, Sebastien Bassong and Roman Pavlyuchenko have all been linked with moves away from White Hart Lane.
Pavlyuchenko is reported to have been the centre of interest from Russian side Lokomotiv Moscow but Redknapp insists he will have to sign a replacement striker before letting the 30-year-old depart.