PARIS (Dow Jones)--Société Générale's board finds itself in an acutely uncomfortable position as French President Nicolas Sarkozy and Finance Minister Christine Lagarde have all but demanded the immediate resignation of chairman and chief executive Daniel Bouton.

The bank faces the embarrassment of having to agree to its sale or breakup - to the chagrin of many in Paris' political and business circles - or risk withering on the vine as customers and staff trickle away from an institution trying to fix its reputation.