Question posted in 
https://core.telegram.org/bots clearly says:
Bots can’t initiate conversations with users. A user must either add them to a group or send them a message first.
But if I visit https://telegram.org/blog/login it says
Telegram bots are a powerful communication tool, but until today they couldn’t start a conversation. Even if you wanted them to reach out to you, you had to chat them up first.
With the new login widget, moving from interacting with a website to a conversation on Telegram becomes completely frictionless.
So it is somehow possible to start a conversation via the login widget. But how do I do it?
I get a user id. Is there an api endpoint to initiate a chat with the user?
2
Answers
As you said, telegram clearly says there is no way for bot to start a conversation, the only thing bots can do is sending messages to an old conversation (I mean a user have to start the conversation).
The only way to begin conversing with a Telegram bot is to start the conversation with them yourself. There’s no way for the bot to start a conversation with a user themselves.