The promoted Swans will pay a club record fee of £3.5million for the 25-year old who scored 27 goals for the Hornets last season.
Swansea chairman Huw Jenkins said that Graham will sign a four-year deal with the club as soon as a medical is complete.
"We have agreed all personal terms with Danny," Jenkins told the club's official website.
"He will have a medical and complete all the formalities once he returns from his holiday.
"Danny has also spoken to Brendan (Rodgers) at length. He is thrilled to conclude the deal and is looking forward to the challenge of Premier League football with Swansea.
"He is a player we have admired for quite some time and as a football club we are delighted to get our man.
"Brendan is looking forward to working with him and there is no doubt he will strengthen our team for the new campaign.''