The club undertook a consultation with members after plans were put to clubs for B sides from the Glasgow duo to play in the league next season.
A vote is set to take place on Monday evening.
If passed, the Colts teams would enter the Lowland League for one season and would be unable to win promotion.
For Bonnyrigg, an “overwhelming majority" of 95 per cent have decided to reject the proposals due to a “number of reasons… but the over-riding one was that of sporting integrity”.
The club feel that if Celtic and Rangers Colts teams were to join the pyramid they should do so at the bottom and work their way up.
They fear that if the proposals are accepted it would turn the Lowland League “into one for hire and that can’t possibly correlate with any semblance of either sporting or pyramid integrity”.
The statement said: “It isn’t that long ago that 13 East of Scotland sides and their board decided unanimously not only to allow us and 25 others entry to the East of Scotland League, but to allow us to play at the same level as existing teams and creating the Conferences. If that didn’t happen we wouldn’t have been promoted to the Lowland League at the first attempt as we would have been in (at least) the 2nd tier of EoS football if we were made to join at the bottom and who knows what might have happened then. They are the only league we can hold up as putting sporting integrity ahead of anything else.
“Looking back on our 1st season in the EoS league our supporters remember the hurdles we had to jump through off the field once the team did the business on it in order to join the Lowland League, with the installation of floodlights in three weeks and campaigns on several different fronts before we were eventually admitted. We, alongside Kelty Hearts, Bo’ness United and Caledonian Braves all had to start at the bottom in order to gain our status in the Lowland League and there are over 100 clubs trying to do the same.
"Our members have no problem whatsoever with any B team joining the pyramid and if the names were Hibs and Hearts or Livingston and Hamilton their decision would have been the same, but that they are treated in the same way as every other new joiner and starting at the bottom and allowed to progress up to Tier 6 in line with other current reserve teams in the EoS and SoS leagues.”
The statement added: “It has been suggested that this proposal will raise the profile of the league but it is our contention that it will not be in a good way, with any goodwill with the wider Scottish Football fraternity built up with the lack of promotion places into League 2/cancelled play-off games etc. immediately erased, and could in theory nullify the legitimacy of the league and potentially put in jeopardy any future pyramid play-offs when league placings of teams competing in our league don’t matter.”