The 25-year-old Spaniard has been heavily linked with a move to the Premier League, as he struggles to hit top form at Estadio Jose Alvalade.
According to Portuguese newspaper, O Jogo, Toffees boss David Moyes wants to bring Suarez to Goodison Park this summer, but could face competition from Merseyside rivals, Liverpool for his signature.
Brendan Rodgers has also been credited with an interest in bringing the former Barcelona player to Anfield, as he looks to continue strengthening his squad ahead of the 2013/14 campaign.
Sporting Lisbon are struggling to cope with their crippling debts, and manager Jesualdo Ferreira has been asked to identify players to make available for transfer this summer.
It's not clear if Suarez is on that list, but it's unlikely the capital club would turn down a good offer from Everton or Liverpool to help ease their current financial situation.
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