The Daily Mail says the Ivory Coast striker, 24, was pictured with chairman Huw Jenkins and a Swansea shirt on Tuesday evening, but Jenkins said there were still "some tiny details to sort out".
He said: "It is 99 per cent done. The fee has been agreed and so has just has just about everything else but there are a couple of signatures that need to be added to some pieces of paper."
It is believed that Bony, who scored 31 goals in 30 games last season, has agreed a four-year contract for a fee of £12million, with up to a further £1.75m in add-ons. It smashes the £5.55m paid for Pablo Hernandez last summer.