Boss Steve Bruce is determined to bolster his attacking options next month to ease the burden on Keinan Davis who remains his only fit centre forward.
Villa are currently without Jonathan Kodjia who could miss the rest of the season with an ankle injury.
Scott Hogan (stomach) and Gabby Agbonlahor (calf) are also currently unavailable.
Bruce has limited funds to strengthen his squad due to Financial Fair Play concerns so is looking to utilise the loan market.
And McBurnie is understood to be among his top targets after netting seven goals in five appearances for Swansea’s under-23s.
Meanwhile, Henri Lansbury has revealed he had to stop himself from jumping in with Villa’s fans after he scored his first goal for the club in the draw at Leeds.
“It was fantastic – I wanted to jump in there and celebrate with our away fans,” he said.
“I was close to going in but there was a wall and thought I’d better not having only just got back from a knee injury.
“I’m delighted to be back out there and it felt great to score. It’s something I’ve been waiting for since the moment I joined. To do it in front of the travelling fans was excellent.”