The Second Division outfit are aiming to pip Aberdeen and Wigan Athletic in the race to snare the young hot shot.
Falkirk's board are due to discuss The Robins' latest offer for Miller on Monday and the 20-year-old is due in Bristol to negotiate his personal terms.
The Robins had an original bid of £200,000 rejected by Falkirk and an improved offer of £300,000 is believed to be sufficient to clinch the deal.
With Miller expected at the club early in the week, the ambitious West Country outfit are confident of landing the promising marksman.
"We hoped to have something decided sooner than this but unfortunately Falkirk arrived back very late on Thursday night from their trip to the Czech Republic," chief executive Colin Sexstone told the club's official website.
"If all goes to plan, and it seems so far, then Lee Miller should be in Bristol on Monday and Tuesday to take a walk around the stadium and discuss terms.
"We are just waiting on the Falkirk board meeting set to take place on Monday.
"Hopefully, we should have something to announce early next week as we don't anticipate any problems with Lee's personal terms."
