The man, who lived 300 metres (yards) from where the woman's partly clothed body was found on Thursday, was still under interrogation on Tuesday.
Police had not charged him with the murder and have until Wednesday evening to make a decision on whether to do so. "He is not denying everything but he is not admitting everything either," local police chief Gilles Soulier told a press conference.
Soulier confirmed that the man's DNA had been identified on items found at the murder scene, in scrubland on the outskirts of the popular tourist town of Nimes in southern France.
He said the man, originally from Chatham in the southern English county of Kent, had been placed in custody after proving "evasive" when asked to account for his movements around the time of the murder.
Deputy prosecutor Stephane Bertrand earlier told reporters that officers had noticed marks on his hands that may have been caused by thorn scratches. The victim's body was discovered in a copse covered in bramble bushes.
The handcuffed suspect was taken to his home on Tuesday morning as police carried out a search of the property which Soulier said had led to the discovery of "quite a few objects that could be linked to the scene of the crime."
A female neighbour who regularly saw the suspect out walking his dog told AFP she could not believe he could have carried out such a brutal crime. "He was a kind person, a very polite young man."
The victim's body was found on Thursday evening after her children's school alerted her partner that she had failed to pick them up.
A box cutter-style knife and two blood-stained stones were found near the body and are thought to have been the weapons used in a murder that police sources described as a "slaughter."
There was evidence to suggest the victim had been sexually assaulted but not raped, police said.
The victim, Joudia Zimmat, was 33 and had three children, aged 3, 6 and 9. A housewife of Tunisian heritage, she regularly went running in the afternoon on tracks in Courbessac, a peaceful residential part of Nimes, which is famous for its amphitheatre and other Roman remains.
She was the third female jogger to have been killed in France since June 2011 but police have not yet established any link with the other murders.