The offer was placed by a consortium headed by David Low, a self-confessed Celtic fan who played a significant role in Fergus McCann's 1994 takeover of the Glasgow club.
Low claimed in a statement that the offer, reportedly worth £3.5million, was rejected by Hibs chairman Rod Petrie because it "involved no financial consideration in respect of their equity interest and shareholder loan".
But the Hibs board claim the best interests of the club were the main factor for their decision. Kwik-Fit founder Sir Tom Farmer owns about 90 per cent of the club with Petrie owning most of the rest of the shares.
A statement from the club board read: "As it has been for 23 years, the first consideration for any proposal received for the football club is whether it is in the best long-term interests of Hibernian.
"Mr Low was given every courtesy to explain the benefits to the football club of his proposals to an independent expert.
"Potential benefit to the club was put ahead of any personal consideration.
"The advice given to the club was that the proposals set out in heads of terms dated 10 July received from Mr Low's lawyers, and again set out at a subsequent meeting held between Mr Low and the independent adviser, were not in the best interests of the club.
"That point was made specifically in the letter to Mr Low's lawyers rejecting the offer.
"Hopefully now in the run up to the Edinburgh derby the club and supporters can remain focused on the football contest on the pitch."
Low's earlier statement had read: "On 18 June 2014 I made an offer for the shares in Hibernian FC owned by HFC Holdings Limited, Sir Thomas Farmer and Mr Roderick Petrie on behalf of myself and other investors.
"That offer was rejected by Mr Petrie yesterday evening (12 August). The principal reason given by Mr Petrie was that it involved no financial consideration in respect of their equity interest and shareholder loan in Hibernian FC.
"I acknowledge that both gentlemen are entitled to reject any offer they do not consider to be in their best financial interests and I accept their decision.
"However, it is apparent that Mr Petrie's decision has more to do with personal wealth management than any Legacy for Leith and, consequently, their estimation of the value of their investment bears no relation to the reality of the marketplace.
"Regretfully, there will be no sunshine on Leith yet."
A fans' group fronted by former Hibs player Paul Kane, Forever Hibernian, has also been in discussions over obtaining a majority stake in the club.