The fee is understood to be £17m with add-ons taking it up to £20m. McBurnie is expected to become a Sheffield United player on Thursday.
The 23-year-old, who was born in Leeds, began his career at Bradford City in 2013, with the Bantams due a slice of the fee.
Gary Bowyer’s League Two side will receive 15 per cent of any profit on the near £500,000 so far paid by Swansea City.
That means Bradford are set to recoup in the region of £2.5m from their Yorkshire rivals’ initial outlay of £17m.
Capped seven times by Scotland, McBurnie has scored 22 goals in 58 appearances for Swansea.
He also netted nine goals in 17 games in a hugely successful loan spell at Barnsley in 2018.
It will be the fourth time United have broken their transfer record since winning promotion to the Premier League.
QPR midfielder Luke Freeman became the first for an undisclosed fee, believed to be £5m, with the £7m signing of Preston striker Callum Robinson following shortly afterwards.
Bournemouth's Lys Mousset was the third time United have broken their transfer record with a deal worth £10m.