Ban said he was "increasingly alarmed" by events in Crimea, where Moscow-backed forces have taken control of the region, which plans to hold a controversial referendum this week.
The pro-Russian parliament in Crimea has scheduled the referendum for Sunday on whether to split with Ukraine and join Russia. Ukraine and Western powers have declared the referendum illegal under international law.
"As tensions and mistrust are growing, I urge all sides to refrain from hasty actions and provocative rhetoric," Ban said in a statement.
Ban also called on "the relevant authorities to ensure that the human rights of all in Ukraine are respected, with particular attention to the rights and protection of minorities."
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Simonovic is to travel to the eastern Ukrainian city of Kharkiv on Monday before heading to Lviv and the Crimean city of Simferopol later this week. He is due to remain in Ukraine until March 15.
Yatsenyuk may meet Ban and members of the UN Security Council during his visit although the details of his itinerary have not been confirmed.
The UN Security Council was holding a behind closed doors meeting on Ukraine in New York on Monday. It is the fifth meeting of the council on Ukraine in 10 days.