Rajapaksa, 69, said his United People's Freedom Alliance had lost after a close battle with the United National Party (UNP) of Prime Minister Ranil Wickremesinghe.
The former two-time president was quoted in the media reports as saying that he conceded defeat "after a good fight".
Rajapaksa said that his UPFA has won eight districts and Wickremesinghe's UNP has 11 out of a total of 22.
It appears that neither the UNP nor the UPFA would gain simple majority of 113 in the 225-member assembly.
The UNP has gained votes in most polling divisions while the UPFA lost votes in comparison with the presidential election held in January.
The UPFA's big victory margins in January presidential elections, 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.
The final outcome would depend now on the number of districts won by the each major party.
In the north's Tamil districts, the Tamil National Alliance appears to have made a clean sweep.
Early results have shown that TNA would win the Jaffna district with a near 60 per cent of the vote.
President Maithripala Sirisena had vowed not to make Rajapaksa premier even if his UPFA wins a majority.
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.
Election Commissioner Deshapriya said he hope the final party positions would be released by this afternoon.
196 members will be 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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