Credit Suisse had a staff of around 45,000 before it nearly collapsed on investor fears about its solvency, which prompted a massive bailout orchestrated by the Swiss government.
Analysts had warned that huge job losses were likely because of the overlapping activities at two of the world's most important banks.
UBS declined to comment on the reported job cuts.
Combined, the two groups had around 120,000 employees at the end of last year, with 37,000 of them in Switzerland.
The Bloomberg report, citing sources close to the companies, said employees had been told of three coming waves of jobs cuts this year -- the first at end-July, the others in September and October.
UBS chief executive Sergio Ermotti had warned earlier this month that the coming months were likely to be "bumpy", saying the merger would require "waves" of difficult decisions, particularly regarding employment.
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