The Sheffield United star could move on this summer following the Blades’ relegation from the Premier League.
A £20million signing from Swansea City, McBurnie struggled to hit the goal trail with United scoring just 20 goals in 38 games altogether.
According to a report in England, Rangers are interested in the 24-year-old who is valued at £15million.
Such a fee would make him the club’s record signing, currently held by Tore Andre Flo.
McBurnie has always been a passionate Rangers fans despite growing up in Yorkshire.
It was through his dad and his side of the family, who are all Scottish, that he developed an affinity with the Premiership champions.
"I never really had a choice,” he told Open Goal last year. "I was always brought up to be Scottish.
"Every non-uniform day I would be going in with Scotland painted on my face and a Rangers top. That's just my old man – that was the way he was.
"Me and my brother always said we were Scottish and Rangers fans.
"It's just one of them. It's a lot easier for me to be a Rangers fan down here than it would be up there."
McBurnie was dealt a huge blow earlier this month with confirmation he will miss this summer’s Euros.
He played his part during qualifying for Steve Clarke but a fractured metatarsal left him “devastated”.
“Not to be able to go to that tournament with the boys is heartbreaking for me,” he said.