The Zurich-based bank confirmed that it was in "discussions" with BofA but that it was too soon to say whether its bid would be successful.
"Given the early stage of these discussions, the outcome is entirely open," the bank said in a brief statement.
Reports in the financial press late last week indicated that two other Swiss banks were in the running -- UBS and Credit Suisse -- along with American bank Wells Fargo and Royal Bank of Canada.
BofA inherited the Merrill Lynch global private wealth management operation at the height of the subprime crisis in 2008.
It is believed to be keen to offload the unit because it does not create sufficient profits owing to its relatively modest size.
The portfolio deals principally with non-US investments in Europe, Asia and Latin America.
The successful bidder may have to pay more than $3 billion to secure the deal, according to the Financial Times.
Analysts Vontobel said that Julius Bär had cash reserves totalling 1 billion francs and would need to raise funds to finalize the deal.
Tying up the purchase would increase the size of Julius Bär's portfolio by 50 percent from its current base of 178 billion francs, Vontobel added, making it a "positive" development.
While the Swiss bank is keen to make the deal, it must first make peace with the US financial authorities, which are investigating potential tax avoidance by its clients.
Zurich's ZKB bank said that Julius Bär may have to pay a substantial fine to resolve the issue, but it had not made any provision to do so.
"From a strategic point of view a Swiss purchase might make more sense, owing to the better synergies (between Swiss entities)," ZKB added.
The Swiss stock market showed little reaction to the announcement, with Bär shares up 1.72 percent to 33.05 francs compared with a 0.55 percent higher Swiss Market Index at 1025 GMT.