The next phase of work to redevelop the Herbert Art Gallery and Museum in Coventry will go ahead, although the city council will not be paying for it.
A full meeting heard on Tuesday night how the leading Conservative group on the council feels the scheme is not among its top priorities.
But it emerged the Alan Edward Higgs Charity has offered financial support to keep the £14m project on track.
Tony O'Neill, deputy council leader, says he is happy the work can go ahead.
He told BBC Coventry and Warwickshire it was good news that the charity has come forward to support the gallery.
The Herbert is due to get new galleries, a glass facade, and provide a new home for the city's archives.
The Coventry-based charity's clerk to the trustees, Peter Knatchbull-Hugessen, told BBC News that the organisation had agreed to help meet the funding gap, which he said was about £3m.
Most of the £14m needed has already been offered by the Regional Development Fund and Heritage Lottery Fund, he said.
Mr Knatchbull-Hugessen said: "As in a 400 metres race, it's won or lost in the last 20 metres and we're now in the last 20 metres."
He said it was no concern of the charity's that the council - of which the Tories recently gained control - has chosen not to come up with the money itself.
"Politicians get elected with priorities. The electorate choose which parties' policies they prefer," Mr Knatchbull-Hugessen said.
He added that the exact figure the charity would spend on the project was still to be agreed with the council.