You can use the PUT method and pass data that will be included in the body of the request:
let data = {"key":"value"}
$.ajax({
type: 'PUT',
url: 'http://example.com/api',
contentType: 'application/json',
data: JSON.stringify(data), // access in body
}).done(function () {
console.log('SUCCESS');
}).fail(function (msg) {
console.log('FAIL');
}).always(function (msg) {
console.log('ALWAYS');
});