McBurnie is at the very top of the wanted list that manager Chris Wilder gave to the club’s owners this summer.
Wilder is keen to acquire Championship stars he believes can make the step up to the Premier League. The Blades are yet to sign anyone since gaining promotion to the top flight.
Brentford’s Neal Maupay is another of those targets, but it seems that Wilder is more interested in bringing McBurnie to South Yorkshire.
United are yet to start official talks with the Swans over a deal for the 23-year-old. However, new Swansea boss Steve Cooper has admitted he cannot guarantee McBurnie will remain at the Liberty Stadium.
When asked if the striker would be staying, Cooper told Wales Online: “I hope so. It’s hard to come across that (his goals). Like any player, if we get an offer you can’t refuse, you have to discuss that.
“You get young players and put them in the team and they thrive, people will look, that’s normal. Anything is possible.”
Why Sheffield United think they’ll get McBurnie
The Blades aren’t the only top flight club in England interested in the Scotland international. Brighton and Hove Albion are also thought to be keeping tabs.
They have the added bonus of being managed by McBurnie’s former Swans boss Graham Potter as well. Potter was in charge at the South Wales club last season as the striker scored 22 league goals.
The 44-year-old has since made the switch to Brighton and it’s believed he could tempt McBurnie to follow him. As yet the Seagulls have not made an approach, though.
United, meanwhile, are hoping the chance to move back to Yorkshire will be enough to bring McBurnie to Bramall Lane. Despite playing for Scotland, the forward was actually born and raised in Leeds.