The museum closed last October to allow work to upgrade the building and provide more display space.
About half of the project's £66m cost is being funded by the council.
In April 2015, the council provided £5.7m to kick-start the building's revamp, which houses treasures donated to the city by collector Sir William Burrell in 1944.
The Burrell collection consists of more than 8,000 artefacts but fewer than a fifth of them have been on show at any one time.
When it re-opens to visitors in 2020, the basement of the building will become part of the exhibition space, so that 90% of the objects can be viewed by the public.
The Heritage Lottery Fund has given £15m to the project and the UK government has also contributed £5m.
It is hoped a fundraising campaign will raise the rest.
Cllr Archie Graham said: "Sir William's great legacy has been described as the greatest gift a city has ever received and we have a moral duty to ensure it is housed in the finest of buildings."

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13 May 2015 Last updated at 12:41 BST
The Haka means "fierce dance" and was traditionally performed by the Maoris before going into battle.
Nowadays the Haka is performed during special ceremonies.
The New Zealand rugby team also perform the special dance before all their international matches.
Prince Harry is currently on a week-long tour of New Zealand.

The Australian was poised to take the lead from eventual winner Lewis Hamilton's Mercedes only for a slow pit stop because his tyres were not ready to delay him.
The problem followed a mistake costing him a win in Spain two weeks ago.
"I got called in and the tyres weren't ready. I didn't make the call," Ricciardo said.
"The guys were running around like ducks, I don't like being up here and being miserable because I got a podium in F1 and it should be a good day but when it happens two races in a row it is hard to take."
He had earlier told the team over the radio on the slowing-down lap: "Nothing you could say would make that any better. Save it."
Watch: Allan McNish explains Red Bull's "colossal mess-up"
Ricciardo pitted a lap after Hamilton finally made a stop to switch from his 'extreme' wet tyres, which he had used from the start of the race, and had built enough of a margin on that tour to reclaim the lead. But the delay at the stop meant Hamilton blasted past as he rejoined the track.
He added: "I felt it should have been me. If Barcelona wasn't clear, this one was crystal clear. I took Barcelona on the chin and this one I don't know if I can handle it as good as I did.
"This is such a prestigious race. I may win it in the future but this one I will never get back so it hurts a lot more than any other."
As for any explanation from the team, he said: "Right now I don't want to hear it. I just want to get out of here. It will be talked about but now is not the time to do it."
Red Bull motorsport chief Helmut Marko said: "We presented it to Mercedes. Unfortunately, a lot of misunderstanding and not the right communication.
"We will investigate and find out [what happened], but it was a human mistake.
"We will check it, make an investigation, and then we can tell you more. I feel really sorry [for Ricciardo], all we can do is apologise to him."
Race result
Monaco Grand Prix coverage details

The 29-year-old, who played for Keighley Cougars and Wales, died from a cardiac arrest triggered by hereditary heart disease.
He collapsed on the pitch during a match at London Skolars on 3 May and efforts to resuscitate him failed.
His widow, Lizzie, has asked people to wear rugby league shirts for the service at St Mary's Church, Halifax.
Mrs Jones said she wanted the service, which began at noon, to be a celebration of the player's life.
BBC Look North's Heidi Tomlinson said hundreds of fans had gathered outside the church to watch the service on big screens.
A fundraising campaign set up by the RFL Benevolent Fund to support the couple's five-month-old twins has so far raised Â£150,000.
A further £27m for the refurbishment of the Burrell Collection has been approved by Glasgow City Council.
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Prince Harry has taken part in a traditional Maori Haka during his tour of New Zealand.
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Red Bull's Daniel Ricciardo said he felt "screwed" by the pit-stop error that cost him victory in Monaco.
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More than 2,000 people are attending the funeral of rugby league player Danny Jones.