Hello.

I will be grateful if you help me with the article:
"EU finance ministers have finished a/the(?)discussion about the possibility of providing more loans to Greece."
Thank you very much for help.

It would be smoother to write this:

EU finance ministers have finished discussing the possibility of providing more loans to Greece.

Regarding the article use in the sentence you typed, you would use "a" if there were two or more discussions, and they finished one of them. You would use "the" if there were only one discussion, and now they're finished.