The ceasefire came after talks in the eastern city of Donetsk attended by Kuchma, officials from the Organisation for Security and Cooperation in Europe (OSCE), the Russian ambassador and representatives from Donetsk and Luhansk regions, Xinhua reported.
The militants in Donetsk and Luhansk regions agreed to halt fire till June 27, the deadline of a week-long truce declared earlier by Poroshenko.
If both sides observe the ceasefire, "then a normal peace process could start", Kuchma said Monday.
Speaking positively of the talks, Poroshenko's deputy chief of staff, Valeryi Chalyi agreed they were "moving in the right direction".
Russian Ambassador to Ukraine Mikhail Zurabov said that the self-proclaimed Donetsk People's Republic and Luhansk People's Republic, would be willing to continue negotiations so as to achieve a lasting truce and peace, and the launch of an inclusive negotiation process.
Poroshenko Friday unveiled a plan offering unilateral ceasefire, power decentralisation in the country through constitutional reform, early parliamentary elections, and local elections in the crisis-hit regions.
It also includes a provision to enhance the status of the Russian language across the country and provides security guarantees to all Ukrainians regardless of political belief.
Under the initiative, a 10-km buffer zone on the Ukraine-Russia border and a special corridor are to be created to allow insurgents to leave crisis-hit areas safely.
So far, the violence in eastern Ukraine has killed more than 250 people, including two Russian journalists.