I had to chuckle at this question becasue you actually made me look up the rule on this one. I am going to say the answer is that yes a lawyer needs to communicate your zero dollar settlement offer to the other side. The relavant rule (4-1) is reproduced below in its entirety. Doesn't matter if the amount is zero or $1.00, the offer should be communicated.
RULE 4-1.4 COMMUNICATION
(a) Informing Client of Status of Representation. A lawyer shall:
(1) promptly inform the client of any decision or circumstance with respect to which the client's informed consent, as defined in terminology, is required by these rules;
(2) reasonably consult with the client about the means by which the client's objectives are to be accomplished;
(3) keep the client reasonably informed about the status of the matter;
(4) promptly comply with reasonable requests for information; and
(5) consult with the client about any relevant limitation on the lawyer's conduct when the lawyer knows or reasonably should know that the client expects assistance not permitted by the Rules of Professional Conduct or other law.
(b) Duty to Explain Matters to Client. A lawyer shall explain a matter to the extent reasonably necessary to permit the client to make informed decisions regarding the representation.
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