The 25-year-old has turned down the offer of a new contract at Everton, as he is looking for a better opportunity of first team football.
The Toffees already have Nigel Martyn and Richard Wright, which means they are well equipped in the goalkeeping department - although manager David Moyes did want him to stay.
Now around four clubs from The Premiership and Division One are ready to offer the former Tranmere Rovers star a new start.
Leeds, who have recently sold Paul Robinson to Tottenham are now believed to be tracking Simonson - but his agent Lee Payne would not confirm their interest.
"There are a number of clubs interested in Steve," Payne told skysports.com.
"We will look at the offers and then Steve will decide what is best for him."
