The director asked for a report on the consultant, ____ he was told had already resigned.

In this sentence, the clause after the comma — “who he was told had already resigned” — contains the word functioning as the subject of the verb had resigned. Although it follows 'on the consultant,'  which might suggest an object form like whom, the grammatical role within the clause determines the choice. 

Since 'who' performs the action (had already resigned), the correct sentence is:
“The director asked for a report on the consultant, who he was told had already resigned.”

If 'whom' were used, it would incorrectly suggest that the word is receiving an action within the clause, ie, suggest it’s the object of a verb, which it isn’t.