Sergei Lavrov said the humanitarian situation there was “deteriorating”.
Ukraine did not authorize the first convoy, which returned to Russia at the weekend, fearing it carried military equipment for pro-Russia separatists.
According to the Ukrainian officials, a column of armored vehicles crossed from Russia on Monday, sparking heavy clashes.
The crossing was reported close to the south-eastern Ukrainian city of Mariupol.
“The Ukrainian border has been breached by a convoy of several dozen tanks and armoured vehicles,” security spokesman Leonid Matyukhin told AFP.
“The convoy has been stopped by border guards… The battle is ongoing.”
More than 2,000 people have died in recent months in fighting between Ukrainian government forces and the separatists. Some 330,000 people have been displaced.
The Russian and Ukrainian presidents are scheduled to meet in Minsk, Belarus, on Tuesday for talks on the crisis.
Sergei Lavrov said he had sent a note to the Ukrainian foreign ministry on Sunday informing it of the new convoy.
He told a news conference on Monday: “The humanitarian situation is not improving but deteriorating.
“We want to reach an agreement on all conditions for delivering a second convoy by the same route… in the coming days.”
Russia said the first convoy had delivered generators, food and drink.
Ukrainian sources said a column of about 30 armored vehicles had entered from Russia close to the port city of Mariupol on Monday, bearing symbols of the self-styled Donetsk People’s Republic.
Mariupol is in the hands of Ukrainian government forces, who ousted rebels in May.
When asked about a possible Russian incursion, Sergei Lavrov said that “there is enough disinformation”.
Ukraine and Western powers have accused Russia of arming the rebels, charges Moscow has denied.
There have been several previous reports of armored vehicles crossing the Ukrainian border.
Asked about Tuesday’s presidential meeting, Sergei Lavrov said: “We are ready… for any format as long as there is a result.”
He added that Russia wanted “to help Ukrainians agree among themselves”.
Sergei Lavrov also commented on the parading of captured Ukrainian government soldiers by rebels through the centre of Donetsk on Sunday.
Crowds lined the streets chanting “fascists” as the disheveled-looking prisoners walked by.
Sergei Lavrov said this was “nowhere near mistreatment” and that Ukrainian fighters’ actions often amounted to “war crimes”.
“I saw images of that parade and I didn’t see anything close to what could be considered as humiliating,” he said.
The violence in east Ukraine erupted in April when pro-Russian separatists in the Donetsk and Luhansk regions declared independence from Kiev. This followed Russia’s annexation of the Crimean peninsula from Ukraine in March.
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