Ukraine, like Georgia, has applied to join Nato, angering Moscow which sees the country as part of its sphere of influence. Ukraine also hopes to join the European Union.
European officials have suggested that Ukraine, which is divided between those who support the pro-Western president and those who back Tymoshenko and favour closer ties with Russia, could be the next flashpoint for tensions between Russia and the West.
Earlier in the week, Tymoshenko blocked a motion that condemned Russia's actions in South Ossetia and Abkhazia, Georgia's breakaway regions.
Cheney flew to Kiev from Georgia, where he had denounced Russia's recognition of the two separatist provinces as independent, calling the move by Moscow "illegitimate" and a "unilateral attempt" to redraw Georgia's borders by force.
Alexander Pikaev, an analyst from the Carnegie Moscow Centre, said the Kremlin would be closely monitoring Cheney's trip and warned it could prompt more violence.
"The Kremlin is certainly concerned that his visit to Georgia could provide him with a false feeling of American support to the [Mikheil] Saakeshvili [the Georgian president] regime and might provoke him to a new attack against separatist entities which Russia has recently recognised as independent states and [is] obliged to protect them."
Ukraine's political problems have intensified in recent days.
In a live national broadcast the president accused Tymoshenko of "a political and constitutional coup" after her party backed the opposition over a law curbing presidential powers and boosting those of the prime minister.
Yushchenko's supporters then withdrew from the governing coalition and the president threatened to call a snap election.
Tymoshenko, once a close ally of Yushchenko, in turn accused the president of having "destroyed" the governing coalition by pulling out of the ruling alliance with her party after the approval of the legislation.
Cheney is expected to express support for the embattled Yushchenko and the country's efforts to join the Nato military alliance.
Valery Chaly, an analyst at the Razumkov Centre for political and economic research in Kiev, said: "Cheney will try to push Ukraine towards preserving the pro-Western coalition."
But he said the crisis would "strengthen the argument" of sceptics opposed to Ukraine joining Nato and the EU.
A pro-Russian member of parliament called for Vladimir Ogryzkov, Ukraine's foreign minister, to be sacked for allowing a US warship to visit Ukraine's Sevastopol naval base.
|Russia's navy uses Sevastopol under a lease due to expire in 2017 [AFP]
Although located on Ukrainian territory, the port is inhabited by both Russia's Black Sea fleet and Ukraine's navy.
Tymoshenko abstained from a vote last month aimed at imposing restrictions on the movements of Russia's navy in Sevastopol while Moscow was involved in military action against Georgia.
Tymoshenko and Yushchenko were the icons of the 2004 pro-Western Orange Revolution and have each been considered Western-leaning politicians despite persistent and sharp disagreements on domestic political issues.
But Yushchenko's supporters have more recently suggested Tymoshenko is doing the Kremlin's bidding - a charge she firmly denies.