The final cost of the scheme is now expected to be £10.7million rather than the originally planned £10.5m.
The scheme was delayed by years following the discovery of major structural problems inside the Vaughan's Mansion section of the site.
It had initially been due to open in November 2011 but finally opened last week.
The issue will be discussed by Shropshire Council's cabinet meeting on Wednesday.
Finance officer James Walton says in a report to the meeting that extra funding has been found from within the council's capital reserves.
The original £10.5m budget already included an extra £910,000 from a contingency fund and Mr Walton said it was necessary to add another £200,000 to that figure.
"The delay in construction has resulted in a final overall budget requirement for the project of £10.7m," he says. "This represents a two per cent increase in cost. This budget increase has been met from funding already in our capital programme available due to other projects completing under budget and is therefore not adding further financial pressure to the council's capital programme."
Councillor Steve Charmley, cabinet member for visitor economy, said the overspend reflected the complexity of the project. He said: "This is a ground-breaking restoration, involving a large number of highly skilled people who have worked incredibly hard over a long period to make it a reality.
"What began as a restoration project turned into a rescue operation as the level and complexity of work required to the intricate structure of the historic buildings revealed itself. These factors have been taken into consideration."