This is in the works 
Monzo certainly are going to offer overdrafts, there’s more details about what they’ll look like (& a request for feedback) here -
Just to let you know, you will be alerted if you’re spending too fast by this functionality. And once you have transactions from the 1st to the last day of any given month, you’ll see the Pulse Graph which has a forecast displaying the date when your balance will reach 0, based on your current spending pattern.
How would the emergency fund be linked to your Targets? I’d be interested to hear more about how this would work 