The Scotland striker, who hit 22 goals in 42 games for the Swans last season, is on the verge of becoming a Blades player, as Neil Wilder's side beef up their squad ahead of the English Premier League season.
United have already brought in French forward Lys Mousset from Bournemouth but smashed their club record transfer fee to land McBurnie.
His price tag dwarfs the previous transfer record for a Scottish player, which was the £15 million West Brom paid to RB Leipzig for Oliver Burke.
The winger had only joined the German side from Nottingham Forest for £13 million - the previous transfer record for a Scot - 12 months prior.
Childhood Rangers fan McBurnie was born in Leeds, and started his career with Leeds United before joining Bradford City in 2010.
He had two loan spells at Chester in 2015, joining Swansea in the summer. Loan spells followed at Newport, Bristol Rovers and Barnsley before the seven-cap Scotland international had a prolific season last term as the Liberty Stadium outfit finished tenth in a competitive Championship.