The shareholding is currently held by Livingston 5 Ltd.
And the club say the offer is "from a company led by John Ward and members of the operations committee".
The club announced details of the potential purchase as well as boardroom changes on their website following Monday's extraordinary general meeting.
"The offer is also subject to the completion of due diligence by the new company and approval by the necessary football authorities," Livingston said in their statement.
"The conclusion of the transaction is due shortly where thereafter the new company will be setting out its future plans which will include a fans share offer with funds raised being ring fenced for the development of the first team playing squad."
The West Lothian club secured their return to the Championship earlier this month, having been relegated from the second tier via the play-offs last season.
The majority shareholders of League One winners Livingston "have accepted, in principle," an offer for the purchase of their stake in the club.