Sigurdsson, 27, made his desire to leave south Wales for Merseyside clear earlier this summer but a deal to bring him to Goodison Park was only agreed on Tuesday evening (15 August).
Koeman's interest in the Iceland international has been well documented in recent months, and the Dutchman is pleased to bring in a player he believes to be vital to Everton's chances of breaking into the top six this season.
"He had his medical this morning and what i heard from the doctor was there was no problem," Koeman said in his press conference. "I don't know if he still needs to sign, but I saw him in the blue, we'll do the business today. Normally he will come.
"We worked a long time to get this deal in and he was one of the key players to bring in. We knew we will lose Lukaku this season and we need players who will bring productivity to the team. We need more players than one to get that number of goals and he is that type of player with a lot of productivity. He had some good seasons in Swansea and is one of best in the Premier League in his position.
Sigurdsson has not had much in the way of a pre-season this summer due to the speculation that has surrounded his future, but Koeman has not ruled out including the Iceland star in his squad to face Manchester City on Monday.
Everton's Europa League qualifier against Hajduk Split on Thursday has come too soon for Sigurdsson, though, and Koeman will talk with his new marquee recruit regarding his fitness.
"I will talk to Gylfi this afternoon," Koeman added. "He will not be involved tomorrow, he did not train today. Maybe for Monday, I need to talk to him and see what his physical state is. Let's wait.
There is an agreement yes, but I don't know if the paperwork is done.
Koeman is pleased that the Sigurdsson saga is finally over, but he told of his discontent at seeing Gareth Barry depart for West Bromwich Albion. The Barcelona legend admitted that he did not want to lose Barry ahead of what could be a gruelling campaign but could not stand in the way of the veteran, who still has a burning desire to play regular football.
"It was a difficult decision when Gareth came in and asked to leave because he knew that he wasn't going to play every weekend and he's still that player who's in love with football," Koeman said. "In the first meeting, I wasn't happy because I wanted to keep him in the team because we expect a tough season, but our relationship was on that level that I gave him the chance to go."