"We are coming to an important phase of the negotiations. We should have, I hope, a deal in the next three to six months," Collardi, chief executive of the private bank, told the daily Le Temps.
He added that both parties are "not at war, but there are fundamental differences in opinion, interpretation and approach to regularise the past. It's a process that takes time."
US tax authorities have been putting pressure on Swiss banks to release information about their American clients. Julius Bär is among 11 Swiss banks currently under investigation for allegedly helping Americans to evade taxes.
Collardi was meanwhile sceptical about calls by the Swiss export industry on the Swiss central bank to raise a floor for the franc against the euro.
Asked whether it would be better if the central bank were to weaken the franc further to 1.30 against the euro, rather than the current 1.20, Collardi said: "Not really."
"It's not so simple. To what level should we raise the peg? In the short term, it's possible and it could make life easier for everyone.
"But in the medium term, maybe it is necessary to adapt to a rate of 1.20 or 1.25 and to adjust our competitiveness. I'm divided on this," he said.
In September, the central bank set a floor of 1.20 for the Swiss franc against the euro in order to shed the Swiss currency's safe haven status which was biting into exporters' earnings.