Supporters of Martin Fayulu say Mr Tshisekedi made a power-sharing deal with outgoing President Joseph Kabila.
Mr Tshisekedi's team denies this.
Final results are due on Friday
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"There were serious doubts on the conformity of the provisional results, as proclaimed by the National Independent Electoral Commission, with the votes cast," it read.
"Accordingly, the [AU] called for the suspension of the proclamation of the final results of the elections."
What might the electoral commission do?
If confirmed, the election result would create the first orderly transfer of power since DR Congo's independence from Belgium in 1960.
The electoral commission said Mr Tshisekedi had received 38.5% of the vote, compared to 34.7% for Mr Fayulu. Ruling coalition candidate Emmanuel Shadary took 23.8%.
However, Mr Fayulu alleges that Mr Tshisekedi made a deal with President Kabila, who has been in office for 18 years.
Mr Fayulu filed an appeal in the Constitutional Court on Saturday demanding a manual recount.
A verdict is expected as early as Friday, and experts say there are three possible outcomes.
The court could confirm Mr Tshisekedi's victory, order a recount, or scrap the results altogether and call fresh elections.
But the court has never overturned results before, and some think most of its judges are close to the governing party.
What do others say?The declaration of Mr Tshisekedi as winner has also been disputed by the influential Catholic Church which says it deployed 40,000 election monitors across the country.
International experts based in the US, and the French and German governments, have also raised doubts.
Meanwhile, the UN says ethnic violence in the west of the country left at least 890 people dead over just three days last month.
Clashes between the Banunu and Batende communities took place in four villages in the area of Yumbi between 16-18 December, according to the UN Human Rights Office.
Voting in the presidential election was postponed in Yumbi because of the violence.
Most of the area's population has reportedly been displaced, including some 16,000 people who sought refuge by crossing the Congo river into neighbouring Republic of Congo, also known as Congo-Brazzaville.
Some 465 houses and buildings were burned down or pillaged, including two primary schools, a health centre, and the office of the independent electoral commission, the UN said.