The opening of the new museum has been delayed for more than two years due to structural problems at the site uncovered during the construction process, with no opening date yet confirmed.
But Shropshire Council's head of financial advice James Walton said the indications are that the project will be delivered within its budget.
The council has earmarked £4.1 million to spend on the Music Hall redevelopment this financial year as part of the £10.5 million scheme.
A statement by the council said: "The £4 million commitment is part of the total scheme budget and relates to the anticipated payment to the main contractor for the planned works on the Music Hall."
The money has been moved into this year's budget due to the hold-ups the project has previously suffered.
In December last year, a Freedom of Information request by the Shropshire Star revealed the council was in talks with contractor ISG Construction over additional costs for the scheme, with delays removing scaffolding believed to have sent bills spiralling.
But Mr Walton said: "I don't know whether that has been concluded. My understanding is we won't go outside the current budget that has been allowed."
The FoI response last December said details of the negotiations between the council and ISG were "commercially sensitive".
But it added "there will be additional costs to the project for the extended period the scaffolding is in situ".
The transformation of the landmark building into a museum and art gallery should have been completed by late 2011 but is running two years behind schedule.
Delays were caused by structural problems being discovered inside the Vaughan's Mansion section of the site.
Restoration work on the multi-million pound scheme first started in the summer of 2009.