Gudjohnsen has been on loan at White Hart Lane from AS Monaco since January and has impressed during his stint at Spurs.
The 32-year-old played 11 times this year, his third spell with a Premier League team having previously played for Bolton and Chelsea, scoring twice as Spurs reached the FA Cup semi-finals and sealed fourth spot and a Champions League qualifying spot for next season.
Tottenham boss Harry Redknapp praised Gudjohnsen for his link up play but reports claim that Martin O'Neill will attempt to take the forward to Villa Park.
The Icelander's father and agent, Arnor, told Sport.co.uk: "It is difficult to make a clear comment at the moment. Firstly, they want to keep him at Tottenham Hotspur. I saw the Aston Villa speculation but I haven't heard from them to be honest.
"He has enjoyed it under the manager. It was difficult for Eidur to come in to the club in the middle of the season, so I think he would like to have a good pre-season with the club and then play a full season with the club.
"I don't know exactly what type of deal Eidur would sign, I wouldn't be surprised if it was a one-year loan deal. But you never know, this has not been decided."
By Owen Edwards
'Obviously Messi is the best player in history and being compared to him means that you're doing things right, but I try to be myself': Barcelona star responds to comparisons with Argentine legend
'Darwin Nunez would be gold for Arsenal': Liverpool legend claims Gunners should have signed Uruguayan striker in January
