Kublai Khan, grandson of Genghis Khan and emperor of the Yuan Dynasty in China, became interested in Japan after he had conquered Korea in 1259. Kublai wanted to rule all of East Asia, which included Japan. At first, an envoy was sent to the Hōjō Regent, Tokimune. The envoy carried a letter that essentially proclaimed the might of the Mongol Empire and it explained that the Koreans had become subjects of the Mongols of their own accord. The letter also stated, in delicate terms, that any answer other than submission to Kublai Khan and his rule would end in a large-scale military conflict.

The letter was forwarded from Dazaifu, the port on Kyushu, to Kamakura. Eventually the letter ended up in Kyoto at the imperial court. Records show that the letter was addressed to the "King of Japan," which we can reason would be meant for the emperor of Japan. The letter was bounced from Hōjō Tokimune to the Emperor, who says he doesn't have the authority to deal with it. So, the letter ends up in Hōjō Tokimune's hands and it is ignored. Kublai sends several more envoys with letters that go unanswered. This leads to the eventual failed invasion by the Mongols and subsequent second failed invasion by the Mongols.

We can assume that the interest was probably politically motivated in the beginning, but the implication of impending conflict was present from the start. I wouldn't call the letters/envoys sent by Kublai Khan negotiations, nor would I say that Kublai was willing to negotiate if the Shogunate had responded in kind.

I know this response is not particularly long, but I hope it answers the question sufficiently.

Sources:

Khubilai Khan: His Life and Times by Morris Rossabi

The History of the Mongol Conquests by J. J. Saunders

Japan Emerging: Premodern History to 1850 ed. by Karl F. Friday

Japan: A Modern History by James L. McCalin

Lecture Notes from: Age of the Samurai