Converse adverbs are provided in this article. The words listed here are commonly found along with the verb converse in sentences. This reference page helps answer the question what are some adverbs that describe or modify the verb CONVERSE.
actually, easily
familiarly, fluently
freely, frequently
intelligently, privately
Hope this word list had the converse adverb you were looking for. In addition to adverbs used with converse, other example verb adverb pairings can be found on this site.
You can help by adding more adverbs in the comments. Thanks.