Rangers chairman Dave King hopes to raise £6 million with the offer.
But Club 1872 – who are the second largest shareholder in Rangers International Football Club plc with a 10.7 per cent stake – face having their shareholding watered down unless they decide to invest.
The group has £960,000 in its accounts but now needs the backing of its members to invest the entire sum in the share issue.
A statement released by the fans group said: “Club 1872 has now been formally invited to participate in the upcoming share issue of RIFC and asked to confirm the level of our investment in this issue by close of business on 14 June.
“The total current funding available to the organisation, across both shares and projects, is approximately £960,000. Club 1872 members are now being polled on whether they wish to designate the share issue as a project – in order to be able to utilise this full sum for the share issue.
“Once we know the outcome of this poll and the level of the additional funds raised this week, Club 1872 will confirm its level of investment to RIFC on 14 June.”
Club 1872 director Laura Fawkes added: “We now have only seven days to confirm the level of our investment and we hope the Rangers support will respond over the coming days. This is a fantastic opportunity for supporters to invest directly into the club and see those funds help to build Steven Gerrard’s playing squad for next season.
“In addition it is an absolutely vital investment for Club 1872 to continue to maintain or grow our shareholding to secure the club for future generations.”