This does not seem to be an option for the hosted password reset page at this time. It looks like it has been discussed, but can’t confirm if/when this would happen.
Sorry for any inconvenience. Feel free to leave us some feedback if this is something you would be interested in. We use this info to gauge customer demand.
Current password on a password reset request doesn’t make much sense (you’ve forgotten your password after all!) but a flow to change the password where you do know the existing definitely would be nice.
Not explicitly for the purpose you are describing. With that being said, you could workaround this by using the resource owner password grant. Hitting this info with the correct credentials will return a token, which you could check for as validation.
Be sure to look at the recommendations for implementing this grant securely.