The Sunday Mirror says Sigurdsson will choose Liverpool or Tottenham Hotspur in the next 24 hours, according to his father Sigurdur Adalsteinsson.
Adalsteinsson said: “Gylfi wants to play at the highest level possible – every player wants to do that – but the opportunity has to be right.
“He hasn’t made up his mind yet. There are plenty of clubs interested in him. There are some in Germany, but Gylfi wants to play in England.
“The main aim for Gylfi is to be in as good a team as he can be and play for the best club possible.
“He is no different to anyone else.
“Liverpool and Tottenham are both in Europe next season so that is not the question.
“I have seen the stories that he is demanding a certain amount of money per week and they are rubbish. He just wants a normal salary and to be on the same as other players at the club.
“The bottom line is he just wants to play and the main thing football-wise is to find the right team which will suit him the most.
“It always helps when you know the coach like Brendan Rodgers. He is a very good trainer. But does that mean Liverpool? Maybe, maybe not.
“He is thinking what is best to do and will have a decision by Sunday.”