Poland's Law and Justice party, which narrowly won general elections last month, is holding coalition talks after its candidate secured the presidency.
Lech Kaczynski has urged Civic Platform to conclude coalition talks
Conservative Lech Kaczynski will be the new Polish president after defeating the Civic Platform's Donald Tusk.
The final results showed he had polled 54%, compared to Mr Tusk's 46%.
Correspondent says forming a government will not be easy as Mr Kaczynski had wooed far-right voters and labelled Civic Platform a dangerous experiment.
Coalition talks between Law and Justice and Civic Platform stalled during the presidential election campaign, which went to the second, run-off vote on Sunday.
Mr Kaczynski said his opponent fought "a splendid battle" and called on Mr Tusk's party to conclude talks on a forming parliamentary coalition.
The conservative prime minister-designate Kazimierz Marcinkiewicz has said he hopes to conclude the talks by Wednesday and to form the cabinet on Saturday.
Turnout was 51%, slightly higher than the first round a fortnight ago, when Mr Tusk was narrowly ahead.
Mr Kaczynski, whose identical twin Jaroslaw heads the Law and Justice party, has been advocating a "moral renewal" and a return to Christian values.
The BBC's Adam Easton, in Warsaw, says the result leaves the Law and Justice party holding the two top jobs in the country.
Both Mr Kaczynski and his party criticised Civic Platform's free market approach, calling it a "dangerous liberal experiment".
The parties also have different visions of Poland's place in the world. Mr Tusk's Civic Platform sees EU membership as an opportunity and they want a speedy adoption of the euro.
But Law and Justice are more wary of Brussels and want to delay getting rid of the Polish currency, the zloty.
Mr Kaczynski's nationalist Catholic beliefs make him more combative towards neighbours and trading partners, Germany and Russia. But the two sides agree on much.
Mr Kaczynski and Mr Tusk are former activists of the Solidarity trade union, which led the country away from Communism.
But our correspondent says that if the sometimes sharp exchanges that characterised the presidential campaign were not just electioneering, coalition talks could be longer and more difficult than many people thought.