This was following a reported request by President Maithripala Sirisena to the UNP leader to step down as premier.
Sources said Wickremesinghe has come under pressure from within the party to handover the party leadership to someone else.
Former president Mahinda Rajapaksa's new party - Sri Lanka People's Party (SLPP) - defeated Sirisena's Sri Lanka Freedom Party (SLFP) and the UNP, winning 225 councils or two-thirds of the 340 councils which went to the polls on Saturday.
Rajapaksa's SLPP swept the polls in a massive wave of support to the former strongman.
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Sirisena's SLFP suffered its worst drubbing with just 13 per cent of the vote.
UNP sources said Sirisena's request to Wickremesinghe to step down was unfair given that the president's party had fared much worse.
"We will form our own government independent of the president's party," a UNP backbencher said.
The next parliamentary election is due only in August, 2020 and the current government's term will run until then.
Rajapaksa, buoyed by the unexpected success, called for immediate dissolution of parliament claiming both Sirisena and Wickremesinghe had lost their 2015 mandate.