Video report by ITV News Correspondent Peter Smith
Administrators are due to be called into Harland and Wolff.
Around 130 people will lose their jobs if the yard closes on Monday evening.
The shipyard has been seeking a new buyer after Harland and Wolff's Norwegian parent company fell into financial difficulty.
The comapny was founded in 1862, by Edward James Harland and Gustav Wilhelm Wolff.
At its height in the early 20th century, the shipyard became one of the biggest ship builders in the world and was Belfast's biggest employer.
Now, around 130 people are employed at the site which now specialises in wind energy and marine engineering projects.
A delegation spoke to a packed chamber which included fellow workers in the public gallery.
At the meeting, councillors unanimously agreed to call on the Government to provide interim funding to allow the yard breathing space to secure its long-term future.
They also called for an emergency meeting with newly appointed Secretary of State Julian Smith.
A banner bearing the “Save Our Shipyard” message has also been hung from one of the landmark cranes, Samson and Goliath, that dominate the city’s skyline.