First-round bids for the Singapore-based unit were submitted last week, sources familiar with events said.
The private wealth business has been valued at around $600m (£414.1m, €530.3m).
DBS, OCBC and Julius Baer declined to comment. Barclays also declined to comment.
The sale is part of a restructuring drive under Barclays’ new chief executive Jes Staley and comes as several European banks rethink their Asian strategy due to pressure to cut costs.
Sources familiar with the sales process also told Reuters that Credit Suisse was weighing a bid for Barclays’ wealth unit, but it wasn’t immediately clear if the Swiss bank had submitted an offer.
Credit Suisse also declined to comment.
Asian wealth grab
Barclays’ Asia wealth business, with offices in Hong Kong, India, Japan and Singapore, had $36bn in assets under management at the end of 2014, according to Asian Private Banker data.
The two Singapore-based bidders are vying for a bigger share of an Asian wealth-management industry that is expanding quickly as rich Chinese gain more and more access to global markets, reports Bloomberg.
Wealth managers with less than $20bn under management in Asia may find it hard to sustain operations and need to consolidate, said Bahren Shaari, chief executive of OCBC’s private banking arm Bank of Singapore, in September last year.