Thank you for your answer, Nicolas!

Yes, but how can I access the request right before the decorated client sends it? Here are some code:

class SigningHttpClient implements HttpClientInterface
{
    use DecoratorTrait;

    // ...

    public function request(string $method, string $url, array $options = []): ResponseInterface
    {
        // I can get these from options, but the decorated client does a lot of logic
        // based on the options and add new headers, creates the body from json, ...
        $headers = $options['headers'] ?? [];
        $body = $options['body'] ?? '';

        // this service needs all the headers and the final body of the request
        $signatures = $this->signer->sign($options['extra']['secret_key'], $method, $url, $headers, $body);
        unset($options['extra']['secret_key']);
        $options['headers'] = [...$options['headers'], ...$signatures];

        return $this->client->request($method, $url, $options);
    }
}

Maybe I could use prepareRequest and other methods from HttpClientTrait, but AFAIK the different client implementations have logic that can modify the request before they actually sending it and in that case the signature will be wrong. Can I somehow get the final headers and body just before the decorated client sends it, without copying a client implementation, or it is not possible? I was wondering that decoration is not a good pattern for this use case and maybe I should use the response and AsyncResponse somehow, but that didn't lead anywhere either.

You need to ensure that your decorator is registered as the last one - just around the real client. Whatever the way to plug into that stack (middleware, listeners, decorators), the priority will always be something you'll need to configure.
Then, yes, the real client can also add some headers. But it won't add them if you already defined them before. If you need to sign only a subset of the headers, then that's going to be easier.
Note that this requirement for signature is strange. HTTPS is a way better way to add security at this layer...