The U.S. Justice Department is pushing BNP Paribas SA to pay more than $10 billion to resolve a criminal probe into allegations it evaded U.S. sanctions against Iran and other countries for years, which would represent one of the largest penalties ever imposed on a bank, according to people familiar with the negotiations.

A final resolution of the yearslong investigation of the French bank is likely weeks away, and it is possible the ultimate settlement amount could total far less than $10 billion. BNP is looking to pay…