They have not been seen since 17 May after visiting their home country of Bangladesh.
It is believed the family stopped in Turkey on their way home before entering the war-torn country.
A statement from relatives in the UK said they were "devastated" and the disappearance was "out of character".
Bedfordshire Police said it had not confirmed whether the family had crossed the border, but inquiries were ongoing. It has not said whether any family member was on any terror lists.
The family is:
The BBC understands Muhammed Mannan has diabetes and his wife Minera Khatun has cancer.
The couple live in Luton with other family members.
The family was reported missing to police by two sons, understood to be Mr Mannan's from a previous marriage, who live in Luton.
One of the sons declined to talk to the BBC.
Police said the family flew to Bangladesh on 10 April and flew back to Istanbul on 11 May. They were due back at Heathrow three days later but failed to return.
Community leader Ashuk Ahmed said he had known the family for 35 years, with Mr Mannan going to the local mosque a number of times a day.
"Initially they went to Bangladesh and on the way back they stopped at Turkey," he said.
"Many families do - Turkish Airlines offers a break in Turkey either to Bangladesh or on the way back.
"The community is extremely worried about the family. There are three generations."
He said no-one knew if they had joined a specific extremist Islamist group in Syria.
However, he said people in the community had told him some women in the family might have been radicalised and took the rest of the family with them to Bangladesh to avoid detection.
Neighbour Syed Hussain said police visited the family house after they went missing and took out paperwork.
He said he heard the family went to Syria and he "felt sorry" for the elderly couple who "probably don't know anything about it".
Bedfordshire Police said a family member had been in touch with relatives in the UK but they did not say where they were.
So far 42 other British jihadists are thought to be in Syria or Iraq.
The total includes Luton man Abu Aziz who fled the UK after being convicted of an attack on a member of the public.
Luton mother-of-six Runa Khan was jailed last year for inciting terrorism in Syria. It is not known if the missing family knew Aziz or Khan.
In a statement, Bedfordshire Police said: "Police can confirm they are investigating the disappearance of a missing family of 12 from Luton who failed to return home from a holiday to Bangladesh.
"There is a suggestion that the family may have gone to Syria, however police have so far been unable to corroborate that information.
"Police are continuing with their enquiries and are working with relatives who are still in the UK."
A statement from relatives said: "We are devastated by the disappearance of these 12 and are very concerned for their safety.
"This is completely out of character and we are very worried of the danger they may now be in.
"This just does not make any sense. We can only think they have been tricked into going there, it is no place for elderly or young people.
"We would urge any of them to please make contact with us or the police to let us know they are ok."
A family of 12 from Luton, including a baby and two grandparents, could have travelled to Syria after going missing, police have said.