Hull City midfielder Sam Clucas is expected to agree personal terms and have a medical on Wednesday.
It is understood that Swansea had an earlier £12m bid for Clucas rejected by Hull.
Lincoln-born Clucas joined Hull from Chesterfield for £1.3m in July 2015 and scored nine goals in 96 appearances.
The 26-year-old was one of Hull's best performers last season when the Tigers were relegated from the Premier League.
Clucas started Hull's first three Sky Bet Championship matches this season, but he did not feature in Saturday's 2-1 defeat at QPR following talk of Swansea interest.
Swansea's need for attacking reinforcements was underlined by head coach Paul Clement following their 4-0 defeat to Manchester United on Saturday.
With Gylfi Sigurdsson sold and last season's top scorer Fernando Llorente out injured, Swansea have managed only one on-target goal attempt in their opening two Premier League games.
As well as Clucas, former Swansea players Wilfried Bony and Joe Allen have both 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, although Stoke have rejected an inquiry for Allen and said the Wales midfielder is not for sale.