The two clubs have come to an agreement in the region of £20m for the England international, and he will now travel to London for a medical.
Fulham have beaten competition from a number of Premier League clubs, including Burnley and West Ham.
Mawson was ever-present for the 'Swans' last season, but he was unable to prevent their slide into the Championship.
He was in the England squad for friendlies last March, but was not selected by Gareth Southgate to go to Russia.