With about 30 percent of the voting stations reporting results, Kirchner, wife of the current president, Nestor Kirchner, was leading a large field of candidates with 43 percent of the vote, The New York Times reported. Her two closest competitors each had about 20 percent.
Under Argentine election law, she can avoid a runoff if she wins 45 percent of the total vote or has 40 percent and a 10-point lead.
If Kirchner wins, she will be Argentina's first female president.
Supporters cheered Kirchner as she gave a victory speech. They also chanted "Ole, ole, ole, Nestor, Nestor" when her husband stood to acknowledge the crowd.
During Nestor Kirchner's four years in office, economic growth averaged 8 percent annually, helping to lift Argentina out of a slump. The election was seen as a referendum on his leadership and as a way for the couple to govern Argentina for 12 years, with Nestor expected to run to succeed his wife.