Analysts predict the Zurich-based bank could face a fine of up to $1 billion as part of a US clampdown on tax evaders, which saw Credit Suisse hit with a $2.6-billion penalty in May.
Chief executive Boris Collardi said there was a "high" probability the bank would settle the case by the end of the year.
"We continue to believe that we will find a fair and equitable solution," Collardi told reporters after the bank announced better-than-expected first half results.
Authorities around the world have been clamping down on tax evasion in the wake of the global financial crisis.
On Monday, the OECD said countries had identified an estimated €37 billion of hidden taxes through voluntary tax disclosure by more than half a million taxpayers.
Julius Bär also announced it would take over the Luxembourg operations of Israeli lender Bank Leumi, which itself agreed to pay US authorities close to $300 million last month.
Leumi Private Bank had 5.9 billion francs ($6.6 billion) of assets under management at the end of June.
Julius Bär will pay 10 million francs in cash as part of the deal, the bank said, adding that it expects to transfer over Leumi clients by the end of 2015.
Overall Julius Bär's net profit in the first half of the year rose 56 percent from the level a year earlier to 179 million francs.
The results boosted Julius Bär shares by more than six percent on the Swiss stock market, against a slightly negative trend.