The Paul Murray-led Knights said they were "surprised and disappointed" to learn the latest development in the takeover of the cash-strapped club.
The rival bidders also warned that the complexities of Miller's proposed newco "should not be underestimated".
Sale Sharks rugby club owner Brian Kennedy and the Blue Knights submitted a "revised bid" for Rangers with administrators Duff & Phelps, on Monday.
But Miller was yesterday named as the preferred bidder in the battle to control the Ibrox side.
A spokesman for the Blue Knights, said: "We believe that the complexities of a newco process should not be underestimated and the likely loss of European foot- ball for a minimum period of three years may carry longer-term financial consequences for Rangers and its many supporters.
"Under our CVA proposed transaction, these issues could have been addressed more easily.
"Whilst we also acknowledge that the delivery of Craig Whyte's shares created a degree of uncertainty, our frustration lies in the fact that it is only in recent days that this issue has been seen as an impediment to overcome in enabling us to make an unconditional CVA offer to Duff & Phelps.
"With the end of the current season now very close, the current uncertainty and delay in resolving matters has to be resolved urgently as there are many key decisions that need to be made to ensure that the club can plan properly for next season and beyond."
In a statement released last night, the consortium insisted its members had the "best interests of Rangers Football Club at heart".
The statement read: "Our motives reflect a thorough understanding of the position that Rangers holds in Glasgow and indeed Scottish football.
"We believe that our bid was, and remains, the best bid for Rangers Football Club and its supporters.
"We were therefore surprised and disappointed to be informed earlier today that a preferred, non-exclusive bidder has been appointed which appears to entail the transfer of the assets and Rangers business into a newco structure.
"Whilst we acknowledge that the acceptance of a newco structure would be easier for Duff & Phelps to execute, we were not prepared to follow that course of action."
Rangers fan groups have called for complete transparency from American tycoon Miller on his plans.
Ross Blyth, from Rangers Supporters' Assembly, said: "We're not as excited as we were this time last year, when it was announced Craig Whyte was taking over the club.
"We obviously need a new owner, but the events of the last year have scared us off people that we don't know that much about.
"So far, Bill Miller has said all the right things.
"In his statement he said that Rangers will not go into liquidation, that the history of the club will be preserved, and the club won't spend outwith its means again.
"These are all things that the fans want to hear, but we need to speak to Bill Miller.
"He needs to come to Glasgow and meet with the fans.
"We need more clarity from him about his plans for the future."
Rangers fans have previously protested against Miller's newco plans, arguing they could spell the end of the club, which was formed in 1872.
During Wednesday night's game against Dundee United at Ibrox, pockets of supporters protested against the US tycoon's takeover bid, with banners declaring Truck off Miller and No Thanks To Asset-Stripping Yanks.
Mr Blyth did not rule out other developments, adding: "I wouldn't say this is all finished yet - there could be a few more rumbles yet to come - perhaps we'll see new bidders and I wouldn't rule out the Blue Knights yet either."
