Citibank plans to offer more than $5 billion in cash for the Turkish bank as part of an auction being overseen by Morgan Stanley, according to the Wall Street Journal.
Belgian bank Dexia and Germany's DZ Bank are also understood to have made a move for Finansbank, but Citibank is believed to be planning the highest offer to date.
Finansbank's current market value is around $5.85 billion, so a higher offer than Citigroup's might be needed to clinch the deal to purchase the company.
The bank is the eighth largest in Turkey and National Bank of Greece is also expected to make a bid for Finansbank this week.
Citigroup was barred by the US Federal Reserve one year ago from completing any "significant" mergers or acquisitions over alleged improprieties. The company's last big deal was the purchase of South Korean bank KorAm in 2004.