At home in Iceland, Gylfi Sigurdsson’s face stares down from billboards advertising everything from Pepsi to the Premier League. Arguably his country’s most famous footballer now Eidur Gudjohnsen’s career is winding down, the Swansea City midfielder has a former Miss Iceland on his arm, the biggest brands pursuing him and is not bad at taking free-kicks. Could this be Iceland’s David Beckham? “You can say that if you want,” Sigurdsson says, laughing.

Nicknamed the Ice Man – a moniker Brendan Rodgers, one of his former managers, was particularly fond of – Sigurdsson is coolness personified. He strolls into the interview room an hour and 45 minutes late, which would be a tad annoying but for the fact he is such a nice bloke and has a good excuse – Sigurdsson, in keeping with his reputation, was out on the training field practising.

Sigurdsson talks like he plays – intelligently – although there is one thing that, on the face of it, is not quite in keeping with the 25-year-old’s suave image. On his Wikipedia page, under “Business Interests” it mentions how Sigurdsson is a “prominent member of the Icelandic fishing industry”, having become chairman of a company in his homeland in 2011, and his boats catch lumpfish, cod, wolffish and plaice.

At Swansea it is well known this is total nonsense and, as a result, a subject which is a source of annoyance to Sigurdsson whenever it is mentioned. In which case, why has he not bothered to edit the damn thing? “I’ve got more important things to do!” Sigurdsson says. “I’d rather play golf than go on to Wikipedia!”

Although Sigurdsson was kicking a football as soon as he could walk, he loves golf. He plays off “four or five” and his brother is a professional in Iceland, where he runs a driving range. With that in mind, it is easy to imagine Sigurdsson’s excitement when he spotted Rory McIlroy wandering outside Old Trafford on the opening day of the season with the Claret Jug in his hand. “He had been there presenting the Open,” Sigurdsson says. “I was just waiting for a taxi because I was going back to the train station. All of a sudden Rory walked past with the trophy, so I asked him for a picture.”

While McIlroy was happy to oblige, word has it the world No1 golfer was oblivious to the fact he was posing for a photograph with the player who had just scored Swansea’s winning goal against the team he supports. “Well, Rory said: ‘All the best’,” Sigurdsson says, “so I’m assuming he knew who I was … unless he was just being polite!”

As for fishing, it transpires it is Sigurdsson’s father whose life revolves around what is caught at sea. When Sigurdsson joined Reading at the age of 15, both parents moved to England with him, and his father ran his fishing business in Iceland via a mobile phone in Berkshire. He also found time to offer Reading’s manager some matchday advice on his son.

Following Brian McDermott’s first game in charge, after Rodgers had been sacked, Sigurdsson’s father politely made the point that substituting a player who scores goals is not a good idea. “To be fair, those two have a good relationship, it’s not like my dad would come up to Garry Monk now and say that,” Sigurdsson says, smiling. “Brian was involved when I was in the youth team and I think they knew each other quite well. But it’s quite funny he said that to him.”

Monk, the Swansea manager, was delighted to get his hands on Sigurdsson this summer. Signed as part of the deal that took Michel Vorm and Ben Davies to Tottenham Hotspur, Sigurdsson is back at the club where he enjoyed a successful loan spell during the second half of the 2011-12 season. With two goals and four assists to his name this season, Sigurdsson is doing the business for Swansea again.

“You never know how long it’s going to take for you to get the first goal but it was nice to get it in the first game [against Manchester United], that probably took some pressure off and allowed me to play my own game and relax,” Sigurdsson says. “But the team plays really good attacking football, so it’s easy to get into the flow of it and enjoy it.”

Getting into the flow of it was trickier at Tottenham. Sigurdsson made 58 Premier League appearances for Spurs after joining from Hoffenheim in 2012, but completed a full game on only nine occasions and rarely got the chance to play in his favourite role. “Of course a little bit of frustration,” Sigurdsson says, reflecting on his time at Spurs. “I think every professional player wants to play every game and every minute because you never know when you’re going to get a chance to score, in the first minute or the 90th. I also probably played too much on the left for my liking – I’m not that kind of player who is going to get the ball and run past the full-back. Of course if it happens, you’ve got to do a job for the team, and no problem. But I feel better – and think you get more out of me – playing me through the middle.”

Those who know Sigurdsson well are convinced he has the ability to play at the highest level, but if there is one slight reservation it relates to that situation at Spurs and whether such an affable man is single-minded enough to bang on the manager’s door and ask to play in his best position. Did Sigurdsson do that at Tottenham?

“You have a chat with the manager, I think it’s just the way you do it,” he says. “You can’t go in screaming, I think you’ve got to be doing it the right way, speaking to the manager and asking for what he’s thinking, how he sees you and what position he thinks will get the most out of you. We had a chat. Tim [Sherwood] came in and I played more through the middle, which I enjoyed a lot.”

Sigurdsson, in fairness, speaks with genuine fondness about his time at Spurs. He loved playing with Gareth Bale in his first season, particularly enjoyed the experience of the Europa League and waxes lyrical about the club’s facilities. “An unbelievable training ground. Probably too good,” he says.

At the same time, he could not tolerate another season being a bit-part player. “It was a tough decision to [leave] because Tottenham are a big club, I really enjoyed playing for them, especially in the London derbies and games like that. But once you finish your career, you don’t want to be looking back and thinking: ‘Maybe I spent too much time there sitting on the bench and not playing.’ I had the chance to come here, I really enjoyed it when I was here last time, and I’m playing regularly, so I think it was the right move for me.”

The personnel and the surroundings have changed a little since Sigurdsson was at Swansea before. Monk is the manager and not a team-mate – “That was strange for the first couple of weeks, I’m not going to lie,” Sigurdsson says – and interviews like this one no longer take place in a cafe at a leisure club, which was where the players reported for training in their first season in the Premier League.

“At Virgin Active,” Sigurdsson says, smiling at the memory. “Swansea have come a massive way since then. They’re in the process of finishing off a new training ground and, no disrespect to the players who were here before, I think the size and the strength of the squad is bigger and better. The club is moving on, they’re talking about expanding the stadium, that just shows the ambition.”

The target for this season, Sigurdsson says, should be “a good FA Cup run and sitting somewhere nicely above 10th place”, which feels realistic given that Swansea are fifth going into Saturday’s home game against Newcastle United. “As for me personally, I’ll keep my goals to myself,” Sigurdsson adds. “You can ask me in May whether I achieved them or not.”