The Herald revealed last year how the "Shaw Burns" was unearthed at a provincial auction in England, and judged by an art expert to have been the work of Alexander Nasmyth, the 18th-century Scottish painter.
The owner of the painting, who wishes to remain anonymous, wants to sell it to a Scottish institution, but this week met officers from Dumfries Museum and discussed a potential loan.
The news comes as new research about the painting, believed to the fourth version of a portrait painted in the late 18th century by Nasmyth after he met Burns, is revealed today at the University of Glasgow's Centre for Robert Burns Studies Conference 2014.
The Shaw Burns, titled because it is believed to have once been owned by Sir James Shaw, the Lord Mayor of London from 1805, will be displayed at the conference.
The owner of the painting, based in England, said he has had offers from art dealers to sell the painting, but remains convinced it should have a home in Scotland.
He met officials from Dumfries Museum because the town is where Burns spent the final years before his death in 1796, and a loan deal was agreed in theory.
Catherine Spiller, museums officer at Dumfries and Galloway Council, said: "We think it is a great idea, it would be a wonderful thing to do, but no date has been agreed on yet.
"It was wonderful to see the painting this week and we were amazed to see its condition and its original frame.
"It is such an exciting find and it was good to see it: you always think you know everything there is to know about Burns and then something like this happens and the story becomes even more interesting."
Other versions of the same painting hang in the Scottish National Portrait Gallery in Edinburgh, the Kelvingrove Art Gallery in Glasgow, and the National Portrait Gallery in London.
The Shaw Burns has been dated from between 1800 to 1810 and authenticated by one of the leading scholars of 18th-century Scottish painting, Dr David Mackie of Cambridge University.
The painting - the only one of Burns taken from life that remains in private hands - has also been investigated by writer Jerry Brannigan.
Mr Brannigan and the owner believe the picture, which shows the Bard in an oval mount with trees and a town in the background, was once the property of Sir James Shaw, the Lord Mayor of London from 1805, who was originally from humble beginnings in Kilmarnock.
It has "Shaw" inscribed on its reverse.
Sir James became a key benefactor to the Burns family after the poet's death, raising funds for them, finding jobs for Burns's relatives in the East India Company and enlisting the help of Sir Walter Scott in aiding Jean Armour, the poet's widow, financially.
When Sir James died, in 1843, he made a bequest to leave his paintings to the London Corporation, but this was not honoured: Mr Brannigan's research suggests his collection was likely sold to raise money to pay off debts incurred in the 1830s.
Last year Dr David Mackie said X-rays of the picture show preparatory work to be in the style of Nasmyth, and he is confident it is by the painter.
Today Mr Brannigan will speak about the painting at the conference, detailing the X-ray and infrared images, its "near perfect condition, original frame and original glass."
He said: "I will talk about Sir James Shaw, who I think was the likely recipient of the painting.
"I can find no other Shaw whose story and timeline comes even close to fitting that of Burns or the painting.
"I find it amazing that when there's no mention at all of Shaw's name while Burns was alive, and only the occasional mention after his death, he appears to be the single biggest benefactor of the family providing money, schooling and jobs for Jean and the children and security for the remainder of their lives."
Professor Gerry Carruthers, co-director of the Centre for Burns Studies, will also deliver a lecture on new research of a Robert Burns manuscript recently acquired by the Mitchell Library.