The 22-year-old will undergo a medical this week before the deal can be officially finalised, according to a statement from the club.
"Swansea City can today confirm a transfer fee and personal terms have been agreed for Hoffenheim midfielder Gylfi Sigurdsson," said a statement. "A club record offer of ?6.8million has been accepted by the German club, and the 22-year-old is expected in Swansea later this week to undergo a medical before the move is confirmed."
Sigurdsson spent the last five months of the season on loan with the Swans and that highly-successful spell has led to the club moving to secure him permanently.
The Icelandic midfielder scored seven goals in 18 league appearances and was also named March's Premier League Player of the Month, becoming the first Swansea player to win the award.
Sigurdsson's move will smash the club's record for a transfer fee, previously held by Danny Graham when he completed a ?3.5million switch from Watford last summer.
"The club are delighted to have agreed terms with both player and club following Sigurdsson's hugely successful loan spell last season," the statement added.
Source: PA
Source: PA