The family of six from Slough in the south-east England county of Berkshire were last seen on April 7 and were reported missing on April 16.
Police have appealed for information for Asif Malik, 31, Sara Kiran, 29, their daughter Zoha, seven, and three sons Essa, four, Zakariya, two, and one-year-old Yhaya.
Thames Valley Police acting Deputy Chief Constable John Campbell said: "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.
Officers believe they travelled south through Europe, possibly by train, and passed through Budapest, Hungary, on or around April 12.
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The report comes as international intelligence officials revealed to 'The Sunday Times' that at least 1,600 Britons have travelled to the Syrian and Iraqi battlefields, more than double the number previously reported.
The newspaper reports that these include at least 100 "jihadist brides" who are in Syria to marry fighters and entire families who have joined IS.
Meanwhile, six people arrested by UK police over alleged terror offences after they were stopped at the Turkey-Syria border have been released without charge.
Five people aged between 21 and 47 were arrested earlier this month on suspicion of commission, preparation or instigation of acts of terrorism.
The five were among a group of nine from Rochdale - five adults and four children - detained in Turkey three weeks ago.
However, no evidence was uncovered to suggest any imminent threat to the communities of Rochdale or the UK.