The 26-year-old midfielder has agreed a four-year deal at the Liberty Stadium.
Clucas becomes manager Paul Clement's first signing since Swansea received a record fee from Everton for Gylfi Sigurdsson.
Press Association Sport understands the deal is worth £15million with another potential £1.5m in add-ons.
The £16.5m fee would represent a club record, surpassing the £15.5m Swansea paid Atletico Madrid for Spanish striker Borja Baston in 2016.
Clucas, who spent time at the Glenn Hoddle Academy in Spain, joined Hull from Chesterfield for £1.3m in July 2015 and scored nine goals in 96 appearances.
He was one of Hull's best performers last season when the Tigers were relegated from the Premier League.
Clucas told the official club website: "I've seen a lot of Swansea and one of the main reasons I decided to come here was because of the way they play football. I think it suits me.
"I had a bittersweet season last year with relegation.
"Because of my playing background, I never thought I would play in the Premier League so I relished every game I played.
"Once you've had a taste for the Premier League you always want to go back.
"So when the chance came up for me to do that again at Swansea, I grabbed the opportunity with both hands.''
Clement has said he wants "two or three players" to fill the void left by Sigurdsson's departure.
Former Swansea players Wilfried Bony and Joe Allen have also been high on the club's list of targets.
Bony is surplus to requirements at Manchester City and Swansea are hopeful of concluding a deal for the Ivory Coast striker, although Stoke have rejected an inquiry for Allen and said the Wales midfielder is not for sale.