The development was announced was made in a statement issued by the Honourable Company of Edinburgh Golfers this morning.
It comes six weeks after the club’s committee failed to secure the two-thirds support it required for a proposal to admit women members.
A postal ballot resulted in a vote of 64 per cent in favour of the resolution and 36 per cent against on a turnout of 95 members of members eligible to vote.
The result led the R&A to drop Muirfield from its rota for the Open Championship, where the game’s oldest major has been held 16 times, most recently in 2013.
Now the committee is going to back to the membership at a Special General Meeting to seek permission to hold a fresh postal ballot before the end of the year.
“A substantial majority of our members voted for change and many have voiced their disappointment with the ballot result and with subsequent events,” said club captain Henry Fairweather.
“The Club Committee believes that a clear and decisive vote in favour of admitting women as members is required to enable us to begin the task of restoring the reputation of the Club that has been damaged by the earlier ballot outcome”.
The result of the first vote, which followed a two-year membership review, was labelled by First Minister Nicola Sturgeon as “indefensible” while Culture Minister Fiona Hyslop said “in terms of our international profile it paints a completely distorted picture of modern Scotland”.
Iain Gray, the former Scottish Labour leader, tabled a motion at Holyrood calling on the club to reconsider its stance “at the earliest opportunity”.
The decision was also slammed by top golfers. North Berwick’s Catriona Matthew said she was “embarrassed” by the vote and claimed it painted a “poor image” of Scottish golf.
Four-time major winner Rory McIlroy said: “I think for golf’s image, as we are trying to break out of this stuffy, old image.
“We are trying to become more with the times and trying to do things to make golf faster, make golf cooler, make more people included.”
On the R&A’s decision to drop the East Lothian venue from The Open list, he added: “It’s not right to host the world’s biggest tournament at a place that does not allow women to be members.
“Hopefully Muirfield can see some sense and we can get it back on The Open rota.”
News of the bid for a fresh votes comes in the week when Royal Troon, venue for next month’s Open, is holding a special meeting to propose the introduction of women members.
It takes place on Friday, when a “yes” result is expected on the back of the Ayrshire club receiving support from three-quarters of its membership during recent consultation.
Club captain Martin Cheyne said: “Recently we spoke about the need for our club to reflect the modern society in which we exist.
“I am pleased that a large majority of members who responded to our survey agree and support opening the club to women.”