The Dons have been chasing the 20-year-old for some time and are hoping a raised bid of around £175,000 can prise him away from Falkirk.
The Bairns were denied entry into the SPL despite topping Division One, and Aberdeen hope the promise of quick cash will tempt them to do business.
Falkirk had originally slapped a £500,000 asking price on the player who has attracted interest from the likes of Nottingham Forest, Aston Villa and even Arsenal.
Pittodrie assistant boss Duncan Shearer says that they will asses how far they are on with the bid when the players return for training on Monday.
Should they fail to land Miller, Shearer has confirmed that the club will then move for former Walsall striker David Zdrilic.
"We'll see where we are when the boys come back for training on Monday," said Shearer.
"That may mean getting Zdrilic up so we can have a look at him because we don't want things to drag on."
Australian international Zdrilic has said that he would welcome an approach from Aberdeen, and would consider a switch to Scotland.
