Forrest, 30, appeared before magistrates in Bordeaux yesterday, four days after he was arrested on suspicion of child abduction.
The married maths teacher from Ringmer, East Sussex, travelled to France with 15-year-old pupil Megan Stammers on September 20. A European arrest warrant was issued after they failed to use return ferry tickets three days later.
Forrest, who appeared wearing a T-shirt, checked shirt and trousers, confirmed his name and nationality before agreeing to return to the UK.
The court is expected to give his extradition the green light tomorrow.
Outside court, Forrest’s British lawyer, Phil Smith, said: “Now that he has agreed to be extradited to the UK, we look forward to the full story emerging.
“He is very appreciative of the support of his family,” said Mr Smith, and “concerned about the impact of this on all those concerned”.
Forrest, Megan’s teacher at Bishop Bell C of E School in Eastbourne, was being kept in detention until his removal.
The teenager has already returned to her home town and been reunited with mother Danielle Wilson and stepfather Martin Stammers.
Appearing in the dock flanked by a female interpreter and two security guards, Forrest was asked if he agreed to return to the UK to explain himself to the British court authorities, to which he replied “Yes”.
The teacher, who gave his birthplace as Aberdeen, then signed a three-page document before the two male and one female magistrates.
Forrest and Megan were spotted in Bordeaux by a member of the public who recognised the teacher from television coverage.
The court in the south-western city’s Place de la Republique was packed with French and British media as reporters and cameramen and women crammed into the building ahead of the short hearing.