The Sun says the striker, 28, will quit the Riverside even if Boro beat the drop.
Gareth Southgate paid a club-record £12.7million to sign him from Heerenveen in January last year.
But they will pocket less than half that sum when they off-load the Brazilian to Portugal.
Alves - contracted until the summer of 2012 - is keen to quit after failing to settle in the North East. He hit only 10 goals in 40 league outings.