The offer has been tabled by a consortium, fronted by the Berwick Rangers' Supporters' Club.
And it comes just days after another director, Craig Forsyth, revealed that negotions were on-going with potential buyers 'from Edinburgh.'
In a statement issued by the consortium on Wednesday morning it stated:
"Berwick Rangers are the subject of a takeover bid this week following a decision by one of their own fans' groups to offer to purchase a majority shareholding from the current club chairman Robert Wilson.
"On the field, the team suffered relegation last season, and are currently close to the bottom of the Scottish Third Division. Only five wins in the last 49 games has seen gates start to fall, with a crowd of only 259 witnessing their recent exit from the Scottish Cup.
"However, a lack of confidence in the future stability of the club has led to the current offer being tabled.
"Bill Parkin, chairman of the Supporters' Club said: 'The time is right for some changes to be made to the structure of the club. We really need to get fans involved at all levels, which is critical for small clubs like Berwick Rangers. We have several parties involved in this bid who are really keen to help turn things around.'"
On Monday Forsyth had revealed that negotiations had been going on 'behind the scenes' with potential buyers from Edinburgh.
And he said that if a deal could be concluded quickly the new owners could be in place as early as this weekend.
Last week Forsyth had made a posting on the internet in which he claimed that he, and two of his fellow board members, including chairman Wilson, were prepared to sell their individual and joint shareholding in the club.
He said that if the price was right, and they thought it was in the best interests of Berwick Rangers, they would accept the offer and 'walk away tomorrow.'
In the statement which appeared on the football-based website 'Pie and Bovril,' Forsyth, under the alias 'Bobby Hogg,' wrote:
"I think you will find Robert (Wilson) is more than willing to step down as chairman as I am as a director.
"We are prepared to sell our shares under RCR Holdings for the price that was paid for them.
"Robert also has 49,000 shares in his own name that were bought for £1 a share and I have 7100 that were bought for £5300 and Mr (Robert) Darling has 5000 that were bought for £5000.
"Mr Darling is also owed £5000 for a directors loan.
"On top of this we are personal guarantors for £10,000 each, which is to cover a £50,000 loan currently being repaid over 10 years. (Think we are now into fourth year). So if someone is prepared to cover the above we would walk away tomorrow."
So far the club has failed to make an official announcement on the situation, and Wilson has refused to comment.
But Forsyth, who has been a director for nearly seven years, confirmed on Monday that: "negotiations with a potential buyer (or buyers) 'from Edinburgh' are ongoing and provided a deal can be struck they might even be in place for this weekend's home league game against Albion Rovers."
Wilson refused to confirm or deny any potential sale this week when he said: "I have nothing to say on the matter.
"Craig has said what he has said, and that is up to him. I do not wish to make any comment about my shares, because that is my business. However, what I will say is that the club is in its healthiest financial position since I became a director in 1996.
"If and when I decide to sell my shares and stand down as chairman I will make a statement then."
With regards to any potential sale Wilson added: "We have entered into negotiations before, but when it has come to the nitty-gritty the talks have fallen through.
"Who knows what the future holds? I might still be here in two weeks, two months or maybe even two years. All I know is that whatever I do will be in the best interests of Berwick Rangers."