Wickremasinghe, 66, would take oath in a simple ceremony at the Presidential Secretariat and the cabinet of national government would be appointed later, media reports said.
The UNP appears to have won at least 11 of the 22 electoral districts with the rival UPFA winning 8 districts.
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"Let us together build a civilised society, build a consensual government and create a new country," he added.
The UPFA's big victory margins in January presidential election, however, have not been affected by the UNP's gains or the votes that have gone to the Marxist JVP or the People's Liberation Front.
Although the final result is yet to be announced, election authorities said the UNP will fall short of a simple majority of 113 in the 225-member assembly.
In the north's Tamil districts, the Tamil National Alliance (TNA) appears to have made a clean sweep by winning three districts.
Meanwhile, Rajapaksa, 69, said his UPFA had lost after a close battle with UNP.
The two-time president was quoted in the media reports as saying that he conceded defeat "after a good fight".
President Maithripala Sirisena had vowed not to make Rajapaksa premier even if his UPFA wins a majority.
A group of Sirisena supporters is likely to join a broad- based national unity government led by Wickremesinghe, who was appointed as Premier soon after Sirisena came into power.
Sirisena was Rajapaksa's health minister until he came forward as the opposition unity candidate to challenge the then president last year. He then handed Rajapaksa a shock defeat in the polls.
A total of 196 members have been elected for a five-year term while 29 will be appointed based on the national proportion of votes polled by each party.
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