GENEVA — Swiss bank Julius Baer Group said Monday that it might have to pay a fine to settle allegations it helped American clients cheat on their taxes, but cautioned investors that the final outcome of the case was still uncertain.

Zurich-based Julius Baer is one of at least 11 Swiss banks under pressure from the United States to give up tax-evading American customers and the bankers who aided them.

Last month, Switzerland’s oldest bank, Wegelin & Co, announced it was selling most of its business after it was indicted in the United States for conspiring to help American clients hide more than $1.2 billion from the Internal Revenue Service.

“We’re in ongoing full cooperation and dialogue with the U.S. authorities and we’re confident and committed that we will find a solution,” Julius Baer’s Chief Executive Boris Collardi said during the presentation of the bank’s full-year results Monday.

Collardi said he was unable to predict the size of any fine his bank might face in the United States, and whether Julius Baer too might face legal action.

“We have not made any provision with regard to the U.S. because we can’t reliably assess how much a related payment would be at this stage,” he said.