Brothers Mohammed Yaseen Safi, Mohammed Ebrar Safi and Bilal Safi, aged three, five and six, have not been traced since they were taken from their foster home over a month ago.
Their father, Imran Safi, is accused of attacking their foster mother before driving them away and police are also hunting for a suspected accomplice photographed in the passenger seat.
Commander Bas Javid, of the Metropolitan Police, said: “I can’t imagine how distressing this whole situation has been for the three young boys, who have been taken away from their safe environment, and our efforts to locate Imran Safi continues.
“We are now offering a £20,000 reward for any information that will help us to bring the children home safely.
“We also need the public’s help to identify a man who was the passenger in the red Nissan at the time the children were taken. He may hold valuable information as to where the children and Imran Safi are.”
Mr Safi turned up at their foster home in Coulsdon Road, Croydon, at around 6pm on 20 August as his sons were playing in the garden.
Commander Jon Savell previously told a press conference the children’s foster mother was threatened with a knife and left with minor injuries
Mr Safi is believed to have driven the children away in a red Nissan Qashqai, which has since been recovered by police.
The children had been in foster care for “some time” and were shortly to be formally adopted – a factor police are looking at as a potential motive for the abduction.
Mr Safi is an Afghan national, but has links to Pakistan, and an “all ports alert” was issued to British airports and borders.
Investigators now believe he and the three children have left the UK, but are continuing attempts to trace them.
“Our working theory is that they have left via a clandestine route, but to where?” Mr Javid said. “We remain unsure, and that is why we need people across the country, and abroad to share this appeal for information.
“The focus of this operation is returning these innocent children to the UK which is our top priority. We hope that offering this reward will lead to some significant information being passed to us.”
Police have so far arrested 15 people in connection with the investigation, including in Croydon, Kenley, Ilford and Bristol.
The suspects, aged between 17 and 45,have been bailed or released under investigation as enquiries continue.
The wanted passenger was travelling in the passenger seat of the red Nissan Qashqai, registration PK13 WFO, as it was driven away from the abduction.
He is described as a bearded man and was believed to be wearing a stripy jumper in the car.
Anyone with any information on the whereabouts of Mr Safi or the three children is urged to call the incident room on 07942599374 or contact Crimestoppers anonymously on 0800 555 111.
Any member of the public who sees them is asked to avoid approaching Mr Safi and call police immediately on 999.