The owner of the centre, Belfast-based PBN Properties, is in talks with officials, after retailers at the Sauchiehall Street site were given a week's notice to pack up and leave.
As the Evening Times revealed, the tenants received a letter from InShops chief executive Frederic Bonnet, saying the operator was getting ready to appoint liquidators.
They were landed with the bombshell news last Friday - and told to vacate by tomorrow.
However, it is understood that PBN Properties, which also owns the office space above the centre, wants to let the businesses carry on trading in the Savoy.
Officials from the property group are currently locked in meetings with representatives from InShops.
An industry insider said: "People are working hard behind closed doors and everyone is hopeful that a resolution will be found."
It came as shock waves travelled through the city, after we revealed that the future of the Savoy, one of the city's most popular markets, was in doubt.
Glasgow Kelvin MSP Sandra White said she was concerned for the businesses following other nearby closures.
She is planning to ask the Scottish Government's Cities Strategies what support there is for the Savoy traders, during a parliament session today.
There are around 60 to 70 businesses set up in the centre.
She said: "It is very concerning. In that part of Sauchiehall Street, Henderson the Jewellers shut down recently. No-one wants to see anything else go.
"The Savoy is a place where people love to go, it has a place in Glasgow's heart.
"My own family use it because it's so diverse."
Anderston/City Greens councillor Nina Baker has urged the council to step in to offer help to the small firms.
She said: "This is a real shock out of the blue and I am calling on the council's business support services, and City Property to reach out and see what help the city can offer.
"The Savoy Centre, as with the Barras, is a treasured remnant of the 'real Glasgow character' trading community and many stalls offer items not readily available elsewhere in the city centre."
In the letter received by the traders, dated January 10, InShops chief executive Mr Bonnet said: "Traders and sub-tenants need to vacate the sites prior to the close of business on January 17."
He wrote that InShops Centre Limited and In-Shops Starters Limited had, "suffered from increasing losses and are now unable to generate sufficient cash to allow them to continue."
Accountancy giant KPMG has organised a meeting of creditors to take place on January 21 in Manchester.
Other shopping developments run by the company across the country have also been hit.