* Deal to improve communication, response to supply threats

* EU says could involve third parties

* EU Commissioner says hopes no gas shutoffs this New Year

* Rules out EU support for Ukrainian gas bills

(Adds Piebalgs comment on Turkmenistan, Azerbaijan)

By Katya Golubkova and Conor Sweeney

MOSCOW, Nov 16 (Reuters) - The European Union and Russia on Monday agreed to an "early warning" mechanism to shield Europe from potential energy supply cuts and protect consumers in the event of a repeat of last year's Russia-Ukraine gas dispute.

The agreement requires both sides to notify the other of any likely disruption to supplies of oil, natural gas or electricity and to work together to resolve the problem. Third parties would also be allowed to participate, the European Commission said.