From the partial results, the RFD has so far won around 12 parliamentary seats with a further five going to the APP, one of its partners in the Forces of Democratic Change coalition, a source said.
"In all probability, we will be the largest single party," RFD leader Ahmed Ould Daddah said.
He predicted they would win around 20 deputies. "Our coalition is still open to new members but we believe that we will be able to form a government."
Independent candidates comprise around one-third of the 45 legislators already elected, sources said, handing them an important say in the balance of power.
Daddah said polling had been transparent but complained the proportional representation system discriminated in favour of smaller parties.
Daddah said the RFD would present an appeal to the electoral commission.
Foreign observers have given the polls a clean bill of health, but raised concerns over a lack of voter education which led many people to spoil their ballots inadvertently.
Diplomats say independent candidates, who were encouraged by the military junta to offer an alternative to established parties, were a mixture of dissidents from the former ruling PDRS party, Islamists and genuine independent politicians.
Many moderate Islamists stood as independent candidates or for the Centrist Reformists party.
"As we looked at who the independent candidates were, there really were a smattering of different people," said Charles Twining, charge d'affaires at the US embassy.
"Once the results are out, we are going to have to look at the individual names to see who is who."