The Iceland worldwide was a critical player as the Liberty Stadium side avoided Premier League relegation towards the end of last season and has been consistently linked with a move to Goodison Park this summer.
The Iceland global has been a target for Everton all summer with the Swans having already knocked back a reported £40million bid.
Sigurdsson has not played since Swansea's opening pre-season game against Barnet on July 12, after which he missed the club's tour to the United States as he was "not feeling in the right frame of mind".
'We are close, we have to wait.
Koeman, the Everton manager whose side host Sevilla in a friendly tomorrow, wants Sigurdsson in place by the start of next week.
On securing other signings, he said: 'Everybody knows we are looking for three more players - a left centre-back, left-back who can play in both positions, a winger and a striker.
Everton get their Premier League campaign underway against Stoke City at Goodison Park on Saturday.
Koeman also responded to rumours linking Aaron Lennon with a move to Watford.
"The reason that he wasn't in the squad is that he was not fit to play today", he added. "I don't know about any interest for Aaron". Should the Icelandic worldwide sign with the Toffees, he would be joining Davy Klaassen, Jordan Pickford, Sandro Ramirez and Wayne Rooney as new arrivals at Goodison Park.