In a story on its website headlined "Levy Headed for Victory", The Post said that the MMD had won 81 of the 150 parliamentary seats up for grabs in Thursday's elections.
If confirmed, the outcome would buck the trend of preliminary results which gave the Patriotic Front some two-thirds of the first seats to be declared.
The Post did not however give figures on how it expected Mr Mwanawasa to fare in the presidential contest in which he faces a strong challenge from the charismatic Patriotic Front leader Michael Sata.
The report appeared as the head of the elections commission, Chief Justice Ireen Mambilima, announced that Mr Sata was comfortably ahead of Mr Mwanawasa in the presidential race after results had been confirmed in 26 constituencies.
According to Mambilima, Sata had secured 49 percent of the first 721,923 votes to be counted, while Mr Mwanawasa had won only 28 percent.
The MMD has been in power in Zambia since 1991 and Mr Mwanawasa is seeking a second and final term after an initial five years in the presidency.