Medvedev said talks would be held on the gas issue, but no further details were given.
In January 2009 a payment dispute between Moscow and Kiev caused a cut-off of Russian gas supplies to over a dozen European countries.
Europe, which gets a fifth of its gas needs from Russia via Ukraine, is hoping warmer Moscow-Kiev ties will prevent repeats of price disputes which have led to supply cuts to Europe.
Many analysts believe Kiev's struggling public finances mean Yanukovych must push for change in the long-term gas deal signed in 2009, which made Russian gas more expensive for Ukraine than for most European countries.
However Medvedev said Moscow and Kiev would work to resolve tensions that had built up under Yanukovych's predecessor, Viktor Yushchenko, who angered Russia by seeking to bring Ukraine into the Nato military alliance.
"There are many questions that have been frozen recently in our relations. We have decided to reanimate them," Medvedev said at the meeting in the Kremlin on Friday.
Aside from gas, the future of Russia's Black Sea Fleet was singled out as a key hot-button issue.
The Black Sea Fleet is based in Ukraine's port of Sevastopol under a lease which expires in 2017, and which the Kremlin is keen to extend.
Yanukovych indicated he was open to compromise with Russia on the fleet's future.
"I think that very soon we will have an answer to this question which will satisfy both Ukraine and Russia," the Ukrainian president said.
Yushchenko had insisted the fleet should leave by the 2017 date.
Yanukovych also said he would scrap orders Yushchenko signed in the waning weeks of his presidency which elevated two controversial World War II-era nationalists reviled by Russia to the status of "Heroes of Ukraine".
The decrees unnerved Ukraine's former imperial master Russia and fuelled its distaste for Yushchenko, who pushed his nation towards Nato and sought to shed Moscow's influence.
Yanukovych also met with Vladimir Putin, the Russian prime minister, who urged the visiting president to bring Ukraine into a Moscow-backed customs union comprising Russia, Belarus and Kazakhstan.