"I can confirm that the French priest is in the hands of Mujahideen [fighters] from Jama'atu Ahlis Sunna Lidda'awati wal-Jihad [Boko Haram's Arabic name], who carried out the operation that was co-ordinated with Ansaru," an anonymous source told the AFP news agency on Friday.
Ansaru is a Boko Haram splinter group that has attacked several Western and Nigerian targets. It claimed responsibility for the kidnap and murder of seven international construction workers earlier in 2013.
Last week, the USA named Boko Haram and Ansaru as Foreign Terrorist Organisations, making it a crime for any US national to provide them with material support.
Father Georges Vandenbeusch, 42, was kidnapped on Thursday in Goza, near his parish of Nguetchewe in northern Cameroon, about 30km from the Nigeria border.
According to witnesses and officials in Cameroon, the abduction was carried out at night by between 10 and 20 gunmen.
The attackers, who are said to have spoken English and Hausa, attacked nuns in the compound, asking for money, before heading to Father Vandenbeusch's house.
Northern Cameroon is considered a "red zone" by the French authorities: French nationals are formally discouraged from visiting the area due to terrorist threats and the risk of kidnapping.
In February, seven members of a French family, including four children, were kidnapped in the same region by Boko Haram.
The Moulin-Fournier family was released after two months.
Nigeria and Cameroon share a long border – 1600 km – stretching from the Atlantic Ocean in the South, to Lake Chad in the North. The northern border is mainly the Sahel semi-desert and is particularly porous.
The Nigerian government's offensive against Boko Haram has pushed nearly 40,000 refugees into neighbouring Niger.
Father Vandenbeusch has lived in the area since 2011. Despite warnings from French authorities, he has continued his work, providing assistance to the 10,000 Nigerian refugees who have fled into northern Cameroon.
His kidnapping has raised fears that northern Cameroon has become a second home to Boko Haram's fighters.
A joint military operation has been launched by Nigerian and Cameroonian security forces in an attempt to locate the kidnapped priest.
Meanwhile, another French national detained by Ansaru has escaped.
Engineer Francis Collomp was kidnapped in the northern Nigerian state of Katsina, near the Niger border, in December 2012.