Typescript definition for a class as an object property

I'm putting together a quick .d.ts for a library, but have hit a snag with the following:

class Issuer {
    constructor(metadata) {
        // ...
        const self = this;
        Object.defineProperty(this, 'Client', {
            value: class Client extends BaseClient {
                static get issuer() {
                    return self;
                }
                get issuer() {
                    return this.constructor.issuer;
                }
            },
        });
    }
    // ...
}

This is used from the user facing API as such (with googleIssuer being an instance of Issuer).

const client = new googleIssuer.Client({
    // ...
})

I've tried extending the Issuer class definition with a namespace with the same name, but this doesn't behave as I had hoped.

What would be the simplest way to represent this in my definition?

Using class and namespace merging you get the equivalent of a static class (a class that can be accessed using the class, not an instance of a class):

declare class Issuer {

}

declare namespace Issuer {
    class Client{}
}
declare class GoogleIssuer extends Issuer {
}
let d = new GoogleIssuer.Client() // Ok
let googleIssuer = new GoogleIssuer();
let c = new googleIssuer.Client() // error

What you want is a class definition that is tied to an instance on the class. For this you can just declare the class as a field of the Issuer class:

// Declare the structure of client separately 
declare class Client {

}
declare class Issuer {
    Client: typeof Client; // Declare a field that is of the class type
}
declare class GoogleIssuer extends Issuer {
}
let googleIssuer = new GoogleIssuer();
let c = new googleIssuer.Client() // ok