Leicester City’s £40m bid for the former Tottenham star has also been turned down, according to BBC Sport, while The Daily Star credited West Ham United with interest in the 27-year-old earlier this month.
Sherwood managed Sigurdsson at North London outfit Tottenham, and the former Aston Villa boss has said that the 27-year-old midfielder would be a success at Goodison Park.
“Everton have already made a number of signings, but there’ll be none bigger than Gylfi Sigurdsson,” Sherwood told talkSPORT.
“He affects the game every time he plays, his dead ball specialties are crucial to any team and he’s up there in the top three in the country with his deliveries.
“He’s a great player, a very good player. He’s never a moment’s problem, he’s easy to manage. I think it’s going to be really difficult for Swansea to keep hold of him. Undoubtedly it’s gone far enough that a fee will be agreed and he’ll be an Everton player.”
Sigurdsson is one of the best midfielders in the Premier League and played well for Swansea during the 2016-17 campaign.
The former Spurs star scored nine goals and provided 13 assists in 38 league appearances for the Swans last season.