How to send DELETE with JSON to the REST API using HttpClient

Although it might be late to answer this question but
I’ve faced a similar problem and the following code worked for me.

HttpRequestMessage request = new HttpRequestMessage
{
    Content = new StringContent("[YOUR JSON GOES HERE]", Encoding.UTF8, "application/json"),
    Method = HttpMethod.Delete,
    RequestUri = new Uri("[YOUR URL GOES HERE]")
};
await httpClient.SendAsync(request);

UPDATE on .NET 5

.NET 5 introduced JsonContent. Here is an extension method using JsonContent:

public static async Task<HttpResponseMessage> DeleteAsJsonAsync<TValue>(this HttpClient httpClient, string requestUri, TValue value)
{
    HttpRequestMessage request = new HttpRequestMessage
    {
        Content = JsonContent.Create(value),
        Method = HttpMethod.Delete,
        RequestUri = new Uri(requestUri, UriKind.Relative)
    };
    return await httpClient.SendAsync(request);
}