The report quoted a spokesman for the Irish prime minister as saying that Varadkar had congratulated Johnson on his election victory in the phone conversation.
Both leaders agreed that there was now a significant opportunity to restore the Good Friday Agreement institutions and pledged to work with the Northern Ireland parties to achieve this, said the spokesman.
The Good Friday Agreement is a 1998 deal that brought Northern Ireland Troubles to an end.
They also discussed how to strengthen the bilateral ties between Ireland and Britain, he said, adding that both looked forward to the smooth passage of the Withdrawal Agreement and agreed to stay in close contact in the period ahead.
Northern Ireland is part of Britain. The Northern Ireland Assembly is the region's top legislature which also has the power to appoint its top executive body. The assembly has virtually stopped functioning since January 2017 due to policy disagreements between members of different political parties within the organization.
Earlier on Friday, Varadkar called on all the sides in Northern Ireland to restore its assembly and executive body by the deadline of January 13, 2020.
"We'll be giving this everything between now and January to get the Assembly and Executive up and running," he said.
"If at that point there is no power-sharing restored in Northern Ireland, we're then looking into another assembly election in Northern Ireland, and I can't imagine who would really want that," he said.
