ATHENS—National Bank of Greece SA, the country's largest bank by assets, may be ready to slightly sweeten its offer for Alpha Bank, but isn't prepared to double the premium, as Alpha is demanding, a senior person familiar with the matter said Saturday.

NBG on Friday announced an offer to acquire Greece's third-largest lender by assets in an all-share deal at a premium that reaches nearly 24%, depending on how it is calculated. Hours later, Alpha's board unanimously rejected the offer, citing the "uncertainties of the current...