Russia announced its suspension on Saturday, accusing Ukraine of misusing the safe shipping corridor for an attack on Russian ships in Crimea. Kyiv has dismissed this as a "false pretext" to withdraw.
The Turkey and UN-brokered deal signed in July by Kyiv and Moscow is crucial to easing a global food crisis caused by the war.
In a phone call with Turkish President Recep Tayyip Erdogan Tuesday, Putin wanted Kyiv to give "real guarantees" that it was "not using the humanitarian corridor for military purposes", a Kremlin statement said.
No grain ship movements were planned for Wednesday, the body overseeing the export deal said, although three more grain-loaded cargo ships left Ukrainian ports on Tuesday.
Moscow had warned on Monday it was "more risky, dangerous" to continue the exports without Russia's participation.
Russia is also putting greater pressure on Ukrainians inside the country as recent attacks damaged the country's infrastructure, plunging families and businesses into darkness weeks before winter.
On Tuesday, Kyiv mayor Vitali Klitschko said water and electricity supplies had been "fully restored" in the capital.
But the attacks have "seriously damaged around 40 percent of the entire energy infrastructure" of Ukraine, the presidency said in a statement.
Ukrainian presidential adviser Oleksiy Arestovich said Monday's bombardment was "one of the most massive shellings of our territory by the army of the Russian Federation".
Following the strikes, aerial views showed Kyiv thrown into darkness overnight, with the only lights coming from street traffic.
Monday's shelling had left 80 percent of the capital's consumers without water and 350,000 homes without electricity.
Klitschko warned there would still be planned power cuts in the city "because of the considerable deficit in the power system after the barbaric attacks of the aggressor".
Ukrainian energy operator Ukrenergo said it would limit supplies to all consumers in central and northern regions of the country to "reduce the pressure on the network".
EU commissioner for energy Kadri Simson arrived in Kyiv "to help scale up support to the Ukrainian energy sector", she said on Twitter.
The Ukrainian army said Russia launched 55 cruise missiles Monday, mainly at energy infrastructure.
Russia's defence ministry claimed Tuesday the "massive strikes... significantly disrupted the management and logistics of the Ukrainian armed forces".
Russia has pivoted to systematically attacking the Ukrainian utilities network after setbacks on the battlefield, where its army is facing pushbacks on the eastern and southern fronts.
In the south, Kyiv's forces are preparing for fierce battles to recapture the city of Kherson and its surrounding region.
Kherson is one of four regions, along with Zaporizhzhia, Donetsk and Lugansk, that Moscow claims to have annexed but does not fully control.
New 'Evacuations' From Kherson
Pro-Moscow authorities meanwhile announced tens of thousands more civilians would be "evacuated" from the Russian-occupied southern Ukrainian region of Kherson amid a counter-offensive by Kyiv.
This comes after 70,000 people already left their homes in Kherson, Moscow-installed local authorities said last week.
The Russian-backed leader of the Kherson region, Vladimir Saldo, said on Tuesday new resettlements were being carried out because of the risk of a "massive missile attack" by Ukrainian forces on a local dam.
But Ukraine said Russian "occupiers are carrying out forced displacement of the civilian population".
"Citizens living in premises along the banks of the Dnipro river are being forcibly evicted from their homes," the General Staff of the Ukrainian Armed Forces said on Facebook Tuesday.
A new Ukrainian counter-offensive was taking shape in the east of the country, where the front line has been static for weeks since the liberation of the route between Kharkiv and Lyman, soldiers in the area told AFP on Tuesday.
Turkey has stepped up diplomatic efforts to salvage the grain export deal that was due to be renewed on November 19.
Erdogan told Putin during Tuesday's call that he was "confident" the issue of grain exports from Ukraine could be resolved, according to the Turkish presidency.
The resumption of the deal could only take place after a "thorough investigation into the circumstances of the incident", Putin told Erdogan.
With millions at risk of starvation unless exports continue, French President Emmanuel Macron "denounced" Russia's decision to exit the deal "which again harms global food security" in a call with his Ukrainian counterpart Volodymyr Zelensky Tuesday, his office said.
Zelensky thanked Macron on Twitter for "specific decisions on strengthening Ukraine's defence capabilities. Specific initiatives to restore the destroyed energy infrastructure".
US State Department spokesman Ned Price accused Russia of deciding to let the developing world go hungry.
"Moscow doesn't care if people starve," he told reporters.