Since the Helsinki proposal was announced more than three years ago, the Guggenheim Foundation has tenaciously promoted it despite critical setbacks, including the local city board’s rejection of the project in a narrow vote in 2012 because of cost concerns, particularly a controversial $30 million licensing fee for the privilege of using the Guggenheim name.

Newspaper polls at the time showed that a majority of city residents opposed the project. In the early stages, local artists balked at the idea that the existing Helsinki City Art Museum would be absorbed into the Guggenheim.

That idea was discarded when the Guggenheim returned with a retooled proposal in 2013, responding to some of the criticism. The foundation agreed not to charge the city its licensing fee but rather to help local supporters solicit private donations through a newly formed foundation. The later proposal also increased estimates of annual admission revenues and reduced by half the operation fees for the Guggenheim, to almost $1.4 million a year.

Not all the critics are satisfied. “This is like we are buying a Louis Vuitton bag because it is a famous brand — we need our own brands,” said Kaarin Taipale, a co-author of “In the Shadow of Guggenheim,” a book that analyzed the project. “The financing numbers have raised a lot of questions. They say we have different alternatives, but all are based on spending public money.”

Money is an important issue in Finland, which has been mired in recession for two years and is struggling with a decline in its electronics industry and a weakening of trade with its neighbor Russia. After the Guggenheim revised the proposal, the city board agreed to reconsider its stance but withhold a final decision until after the designs were evaluated.

When the design competition was announced last month, a Finnish television reporter asked Mr. Armstrong what alternatives he might pursue if the city or state refused to pay for the project. “These are typical bad questions,” he replied, and then walked out. Eventually he returned to resume the interview, but it was too late: The video of his remarks spread widely on the Internet, provoking accusations of arrogance.