The Blue Knights group led by ex-Rangers director Paul Murray was "surprised and disappointed" after losing out on the preferred bidder status.
Mr Murray said he still believed the consortium's bid – worth about £5 million – provided the best chance of preserving Rangers' history and heritage.
He also warned that "complexities" over Bill Miller's plans for a Rangers newco should not be underestimated.
The Blue Knights – which also includes Brian Kennedy, the Scots-born owner of the Sale Sharks rugby team – released a statement, saying: "We believe the complexities of a newco process should not be underestimated and the likely loss of European football for a minimum period of three years may carry longer term financial consequences to Rangers Football Club and its many supporters."
The Blue Knights said a Company Voluntary Arrangement (CVA) would have allowed issues surrounding European football to be more easily addressed.
The statement added: "We believe our bid was, and remains, the best bid for Rangers Football Club and its supporters.
"We were therefore surprised and disappointed to be informed 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."
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