Villa have made a very bright start to the season in the Championship. Following their 1-1 draw away at Leeds on Friday night, Steve Bruce's men sit fifth in the second tier and are unbeaten in their last four.
The January window may be a crucial one for Bruce as he looks to try and give the side the best chance to capitalise on their current position, but The Sun on Sunday reports that he may not have the funds to make too many significant signings.
It has been a frustrating season for the 21-year-old at the Liberty Stadium, with just seven substitute appearances coming in the Premier League this term - while cumulatively, he has spent less than 90 minutes on the pitch in the top flight.
With the Swans struggling for goals this season, it may perhaps be a surprise if Paul Clement is particularly keen to let McBurnie go unless he manages to get a new striker in anyway as they look to get themselves out of trouble.
But Villa fans may not be particularly inspired by the links with the Scotland under-21 international - though he scored a brace for Swansea's under-23 side against Sunderland recently, and bagged a hat-trick against their Chelsea counterparts in October.
The Villans arguably need a reliable goalscorer with Jonathan Kodjia and Scott Hogan sidelined with injury, so a move for McBurnie may be a gamble given that all of his eight senior goals to date have come in League Two or the National League.