The Lady In A Fur Wrap was gifted to the city in 1966 but the identity of its painter has long been a topic of debate.
The portrait is currently on loan to Toledo in Spain, as part of the celebrations marking the 400th anniversary of the El Greco's death.
El Greco was the Greek born artist Domenikos Theotokopoulos, who resided for much of his life in Spain.
Researchers at Glasgow University, the Prado museum in Spain, National Media Museum, Bradford and the Centro de Estudios Europa Hispanica, Madrid are now planning to carry out what they hope will be definitive research on the painting, which is usually displayed at Pollok House.
The picture was discovered in Paris more than 200 years after the artist died.
Differing theories have also been offered about the identity of the lady pictured in the painting - there have been attempts to name her as a member of a royal family, or even El Greco's mistress or daughter.
The investigation, as part of the ongoing Stirling Maxwell Project, will be led by Dr Hilary Macartney from Glasgow University, who said that it will be collaborating with Glasgow Museums on a research project on the Spanish paintings in their possession from the former collection of Sir William Stirling Maxwell.
Sir William had close links to the University as he was Chancellor from 1875 to 1878.
A spokesman for Glasgow Life said: "Within the art world, there are many debates between scholars and academics over the provenance of works.
"Lady In A Fur Wrap has been the subject of such debate for decades and we welcome any contribution which will provide more information on this important work."
Glasgow Museums' guide says the El Greco attribution is despite the smooth modelling of her face being "unique in his oeuvre".