Azarov served as prime minister under Ukraine's Moscow-friendly President Viktor Yanukovych, who was forced from office by protesters and fled to Russia in February 2014.
Rubio was speaking after attending the meeting of G7 foreign ministers in Paris where his European counterparts had repeatedly emphasised the importance of being directly involved in the talks.
Rustem Umerov, the head of Ukraine's Foreign Intelligence Service who is leading Kyiv's delegation, thanked the U.S. for its efforts to end Russia's invasion.
