The former Leeds United star has also had interest from The Whites as Marcelo Bielsa looks to strengthen his side for the Premier League.
What We Know
As far as Burnley's interest goes, so far it's only been an Enquiry. However, last week West Ham had a £12m bid rejected as Sampdoria try to keep hold of their star man.
The Clarets have also enquired about SCO Angers midfielder Baptiste Santamaria for a similar price of £12m. Sean Dyche clearly wants to add in that position with the departure of Jeff Hendrick and Jack Cork being on the wrong side of 30.
Where he would fit
Burnley's most recent midfield acquisition is Josh Brownhill, the 24-year-old has shown his ability to become a box-to-box midfielder capable of dominating the middle of the park for Burnley.
However, Brownhill is more attacking than defensive and with Ashley Westwood and Cork both ageing, a long term partner for Brownhill would need to be defensively sound. This would explain the interest in Santamaria and Vieira, two defensive-minded midfielders.
Vieira is just 21-years-old and would be the perfect long term fit for Burnley. The Englishman has already made over 70 appearances for Leeds in the Championship and is used to the physical side of British football.
A partnership with Brownhill would give Dyche two young, talented midfielders whose jobs would work closely together, with one more defensive and one more attacking.
Of course the only real issue with a transfer link like this is the cost, Burnley are known for spending little and making the most of what they've got. However, Dyche has vocalised his displeasure with the lack of ambition from the board and Vieira would be the perfect way to start changing that. If the board do pay out, the club would be a great place for a young talent like Vieira to learn and grow.