The man, identified by family friends as Robert Plant, lived with his mother 300 metres (yards) from where the woman's partly clothed body was found on Thursday.
He remained under interrogation on Tuesday evening but had not been charged with murder. Police have until Wednesday evening to make a decision on whether
to do so.
Police were first directed towards Plant when a composite image – generated from several witness descriptions of a man who had been acting strangely in the area of the murder on January 23rd – matched his own facial characteristics, according to a report in regional daily La Depeche on Wednesday.
"He is not denying everything but he is not admitting everything either," local police chief Gilles Soulier told a press conference.
Soulier said the detained 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 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."
Despite these incriminating signs, one report on Wednesday claimed that Plant's mother had attempted to provide him an alibi for the day of the murder, when questioned by investigators.
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."
A former colleague of Robert Plant's late father told AFP: "They are a lovely, kind family. There is nothing in Robert's background to suggest he might have done something like this."
According to the colleague, the suspect's father Denis had been a telecoms executive who was half-French and had worked for much of his career in Paris before retiring to Nimes, where he died last year. The family was originally from Chatham in Kent.
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.