Father Asif Malik, mother Sara Kiran and four children aged between seven and one were seen boarding a ferry from Dover to Calais before travelling south through Europe by train, according to Thames Valley Police.
Police say they are increasingly concerned about the family, who have not been seen in England since April 8.
There are also sightings of the family, from Slough, Berkshire, in parts of south Europe.
A police spokesman said: “Given the online rhetoric from the Islamic State (ISIL) encouraging people to travel to Syria, the direction of travel and concerns voiced by Asif’s family, we are concerned the family may be seeking to travel to Syria.”
The family left Slough without mentioning any holiday or travel plans to family members, which is entirely out of character for this family, police said.
Mr Malik is 31, his partner is 29, and they have a seven-year-old daughter Malik, and sons Essa, four, Zakariya, two, and Yhaya, one.
Police believe the family, who caught the 12.30am ferry from Dover 11 days ago, might have travelled through Budapest on or around Sunday April 12.
Acting Deputy Chief Constable John Campbell said: “We are extremely concerned for the safety of this family, especially the four young children.
“We are reaching out to Asif and Sara in the hope that they hear our messages, and we encourage them to make contact with us so we can return them safely to their families who are extremely worried about them.
“We would urge anyone with information about the family’s whereabouts to come forward and speak to police so the family can safely return to the UK.
“We would ask anyone who saw the family as they travelled to Dover, waiting at the port, on the ferry or when they arrived at Calais to contact the police.
“Any piece of information, no matter how small, could help the UK or European authorities to locate the family so that they can be safely returned home to their loved ones.”
A police spokesman said there was “no suggestion (the family) were doing any fighting” in Syria, and were unable to confirm whether the family were sympathetic to the rival causes of either Syrian president Bashar al-Assad or ISIL.