The Premier League side want to capture the services of highly-rated Collins and have already failed with one offer to snap him up this month.
Sean Dyche’s side offered Stoke an initial £4.5m for Collins and have now come back with a proposal of £7.5m.
However, Stoke have again turned down the proposal for Collins as they value the defender at a higher price.
No other club have offered as much as Burnley now have for Collins, but the clock is ticking on the transfer window if the Clarets want to try again for the player.
Collins has also been linked with Crystal Palace and Manchester United.
The defender has impressed with his potential at Stoke and also has Arsenal amongst his admirers.
Stoke currently sit tenth in the Championship table and keeping hold of Collins would boost the Potters’ promotion bid.