The Clarets suffered their 16th defeat in 22 Premier League fixtures on Wednesday evening as they lost 3-1 to Manchester City at the Etihad Stadium.
After the game, head coach Vincent Kompany indicated that he was hopeful of signing at least one fresh face before close of business on Thursday.
Kompany openly acknowledged that Montpellier HSC defender Maxime Esteve was specifically a player that he was looking to add to his backline.
However, according to the Daily Mail, Kompany is also attempting to pull off the eye-catching signing of Meunier.
Since 2016, the right-back has represented either Paris Saint-Germain or Dortmund, at one time being regarded as one of the top players in his position in Europe.
Meunier has endured a frustrating 2023-24, however, with just six starts and one substitute outing being made in the Bundesliga, three of which were made in January.
Although the 32-year-old has recently featured in teams that have defeated Darmstadt, FC Koln and Bochum, it appears that Dortmund are open to parting ways with the Belgium international.
Less than six months remain on his contract at the Westfalenstadion with the German giants seemingly prepared to reduce their wage bill before the end of the day.
The same report adds that Burnley may allow Connor Roberts to move to Leeds United if they can acquire the services of Meunier.
Leeds have been in the market for a new right-back ever since they allowed Djed Spence to return to parent club Tottenham Hotspur earlier in the month.
Although the Wales international has made 16 appearances in all competitions this season, he has not played more than 31 minutes in a top-flight fixture since October 23.
With Roberts having 17 months remaining on his contract, it is unclear whether Burnley would insist on an option to buy if they agree to bring a positive conclusion to negotiations.