Leeds have enquired about the availability of Brownhill, who is also being linked with West Ham United and Aston Villa, ahead of the transfer window slamming shut.
The Whites’ interest comes despite boss Marcelo Bielsa insisting he is not expecting any more signings before the deadline.
Jack Harrison has also moved from Manchester City on a permanent basis following a succession of loan deals.
Brownhill left Championship outfit Bristol City for Burnley in a £9million deal last year and has started the Clarets’ last two Premier League outings, including the stalemate with Leeds at the weekend.
Former All Leeds TV anchorman Wainman believes Bielsa is willing to do business for the 25-year-old but is not able to pay a significant fee.
He told Football FanCast: “At the minute, Leeds United aren’t willing to stump up a big figure for players.
“But he is someone I would take because we need a centre midfielder.”