Thousands of people travel to the museum in Pollok Park every year to Sir William's world-renowned art collection, which he gifted to the people of Glasgow.
Under the original terms of Sir William's 1944 Deed of Gift, the exhibits cannot be shown overseas.
Now, 16 years after a costly and controversial battle to change the rules, Glasgow City Council is preparing to launch a parliamentary bid to confirm that art from the 8000-piece collection can be seen by a wider international audience.
The 30-year-old building housing the art requires a major refurbishment, estimated to cost millions of pounds. This will, if the funding is raised, lead to its closure for a revamp, possibly between 2016 and 2020.
Now, with the full backing of the trustees of the Burrell collection, a Private Bill will be put forward in the Scottish Parliament to lift the current restrictions on overseas lending, so the treasures of the collection can tour overseas.
Those behind the plans say the foreign tour would enhance the city and collection's reputation and raise money for the extensive redevelopment.
Papers detailing these proposals will be published today and will be considered by the council next week.
The Bill to change the rules of Sir William's deed of gift is expected in the spring, with, it is understood, support from the Scottish Government.
The council is assessing the state of the building but there is a view the museum, opened to acclaim in 1983, is "not fit for purpose" and requires a major overhaul, especially its roof.
In 1997 the council won a battle to change the Burrell bequest so its treasures could tour overseas. But two vital factors, after a Parliamentary Commission ruling in the council's favour, were not resolved: the need for a parliamentary bill to change the bequest, and the full support of the trustees.
Now, after careful negotiations over years, the trustees, led by Sir Peter Hutchison, as well the council and its arm's-length organisation, Glasgow Life, which runs the city's museums and galleries, are determined to settle the unfinished business.
Sir Angus Grossart, a Glasgow Life board member and leading banker, will be the chairman of the newly formed Burrell Renaissance Council, which will oversee the refurbishment programme and tour.
Sir Peter said: "The trustees welcome the forthcoming refurbishment, which will transform the Burrell building and provide a fitting context for this world-class collection.
"New gallery space will be created, a wider range of objects displayed, facilities upgraded, and any structural defects, such as the roof, remedied.
"The trustees also support the application to the Scottish Parliament to enable a major tour during the period of closure which would not only assist with fundraising but would also raise the Burrell's profile and demonstrate the extraordinary range and quality of Sir William's unique legacy".
Sir William, who died in 1958 aged 96, stipulated he would not allow any works to be loaned overseas. As a shipping magnate, he was conscious of how dangerous transporting art by sea could be. However, all international loans are now transported by air.
The collection has been compared in quality to those held by the V&A and the Metropolitan Museum of New York, especially its Chinese, early Renaissance, French, Islamic, Egyptian art and Greek and Roman antiquities.
The move could bring other benefits, such as widening access to the collection, raising visitor numbers, improving international partnerships with galleries and opening it to international scholarly research.
The Burrell trustees previously varied the condition to require the collection to be housed more than 16 miles from the city centre. However, the trustees agreed to allow the collection to be housed at Pollok Park as pollution was no longer considered an issue after the introduction of the Clean Air Act 1956.
Councillor Archie Graham, depute leader of Glasgow City Council and chairman of Glasgow Life, said: "Sir William Burrell's gift to the people of Glasgow cannot be underestimated. We are working to find a way to make this international touring exhibition happen.
"Sir William entrusted his life's work to the city he called home.
"I am determined any work we progress will be mindful of his wishes and help to secure much wider recognition for his vision and achievements as we look toward protecting the collection's home for decades to come."