From your website, newsletters, or other channels, you can use the URL shortener from Meta (Instagram) to redirect clients to a conversation in your instagram.
When a client starts a new conversation or continues an existing conversation with your Instagram account, using an ig.me link, the client will be redirected to a new/existing conversation with that Instagram account.
To create the link, use the format https://ig.me/m/USERNAME where USERNAME is the username or @ for the Instagram account.
In the example below, an email is sent with an ig.me link, and from a phone you can directly log in and open the conversation:
You can find more information in this article from Meta.
The ig.me link is currently only compatible with the v210 versions of the app and later and does not work from a desktop, just from a mobile device.