Asif Malik, 31, and partner Sara Kiran, 29, vanished with daughter Zoha, seven, and sons Essa, four, Zakariya, two, and 12-month-old Yhaya two weeks ago.
Asif is a former member of the banned radical group Al Muhajiroun and a close associate of its former leader Choudary.
Police said the “direction of travel” and the fact that Asif had told family members he wanted to live in a Muslim country made them fear they were going to Syria.
The Maliks were last seen at their home in Slough, Berks, on April 7. They were caught on CCTV boarding the 12.30am Dover to Calais ferry the next day.
Police believe they then travelled across Europe, by train, going through Budapest on April 12, before heading to Istanbul. From there it is a short bus ride to IS-controlled territory in Syria.
The family were reported missing on Thursday by Asif’s mother.
Choudary described Asif as a “champion for shariah” who regularly attended Al Mujahiroun’s demonstrations.
He said he remained “very influential in the community” after the group was banned for alleged terrorism links five years ago.
Mr Choudary said: “Asif is a very good friend of mine but I haven’t seen him for a while.
“I last saw him a couple of months ago - he used to attend my lectures in Slough and west London.
“He is a very studious man, a very good Muslim character. I’ve known him for about 10 years.”
He denied any knowledge of the family’s plans but criticised Britain’s efforts to stop its citizens travelling to Syria.
“The suggestion that he has gone to join Islamic State is an assumption. You don’t have to tell the whole town before you go.
“No one should be restricted to travel wherever they want. Why are Muslims being restricted?
“I don’t think that people who are travelling abroad with their wives and children are going somewhere to blow themselves up.
“I think they want to live under sharia and escape the oppressive life they have in this country and under this government.”
John Campbell, acting deputy chief constable of Thames Valley Police, said the force was treating the Maliks as “missing persons” and had asked Turkish authorities to help track them down.
He said: “We are extremely concerned for the safety of this family, especially the four young children.
“We’re concerned about anyone intending to travel to the part of Syria controlled by Islamic State. It is an extremely dangerous place and not a place where young children should be taken.”
Neither Asif nor Sara has contacted family or friends since leaving the UK.
In a statement, Asif’s family said: “We are totally distraught and in shock. The greatest sadness is the decision by Asif and Sara to travel with their young family, such beautiful children.”
They appealed to Asif to “think about what has happened and the decisions you have made” and called on him to get in touch to “let us know that you and the children are well”.
A neighbour of the family, who did not want to be named, said officers broke down the front door of the family’s flat on Thursday.
He added: “They were a lovely family. I would see them with the kids every day. They were just really nice people.
“We started hearing rumours that they had vanished a couple of days ago.
“A lot of people were saying that they were extremists and were very, very strong in their beliefs.
“But when I saw them I wouldn’t have thought that because he appeared perfectly normal.
“The thing that surprises me is that they took the kids, who are really young.
“It’s such a shame because these kids are now in danger because of what their parents believe in.
“How extreme and stupid it is that they are putting their kids in danger.”
Anyone with information should call Thames Valley Police on 101 quoting reference number 342 (19/4), or call the Anti-Terrorist Hotline number on 0800 789 321 or from outside the UK on 0044 1865 841 148.