The Herald understands that 50 jobs are likely to go at the operation, although this will not occur until the end of 2013 and some staff are likely to be offered other roles within the company.
Citigroup said it is committed to retaining a base in the capital.
A spokesman told The Herald: "Citi is proposing to move a limited number of roles currently based in Edinburgh to locations outside the UK.
"Citi will work closely with the affected employees, including exploring transitions to other roles within the bank. Citi remains committed to doing business in Scotland.
"The proposed move would allow Citi to improve efficiency while maintaining the level of services provided to our clients."
The decision comes at an uncertain time for Citigroup which unexpectedly parted ways with group chief executive Vikram Pandit this week.
Around 300 people currently work for Citigroup in Edinburgh.
Citi's Edinburgh office was opened in 2003 providing functions such as fund accounting and reporting, derivatives, tax services, data management, performance measurement and client services.
Customers include Standard Life Investments and Kames Capital, the fund management arm of Edinburgh-based insurer Aegon UK.
So-called "offshoring", often to cheaper but English-speaking centres in India, has become common in financial services as companies seek to cut costs.
Royal Bank of Scotland's insurance arm Direct Line recently transferred 400 roles overseas in the run-up to its flotation.
After a meeting with staff on Tuesday, Colin Wilson, managing director of Citi's institutional clients group in Edinburgh, told them in a follow-up email seen by The Herald that some work will be moved to "lower cost locations and to locations that will enable fund services operations to be delivered to clients on a globalised 24 hour basis".
In Edinburgh the future focus will be on more client-facing services.
Further job cuts are being considered "in view of the challenging economic environment and emphasis on tighter expense control", Mr Wilson wrote.
The capital has become a hub for fund administration work with businesses such as State Street and Bank of New York Mellon running operations in the city.
Glasgow is also home to some financial services administration operations. Citigroup itself employs 200 people in the city after acquiring an administration business for "wrap" investments from Scottish Friendly last year. They will not be affected by the changes.
Some analysts anticipate that the departure of Mr Pandit will lead to an increased emphasis on cost-cutting at Citigroup.
Mr Pandit and John Havens, Citigroup's president and chief operating officer, abruptly stepped down reportedly following a clash with the company's board over the bank's strategy and performance.
Citigroup, the US's third-largest bank by assets, named Mike Corbat, previously its London-based chief executive for Europe, the Middle East and Africa, to replace Mr Pandit.