The family of two adults and nine children were told by a US official that their right to travel had been revoked as they waited in the departure lounge of Gatwick airport on last Tuesday.
They say they had been granted travel authorisation online several weeks previously and were given no reason for the last-minute reversal.
Lawmaker Stella Creasy, who represents the north London area where the family lives, said she had asked Cameron's office for help in resolving the case.
She told Sky News: "It is not just the family themselves who are livid. The vacuum created by a refusal to provide any context for these decisions is fuelling resentment and debate."
She said there was a growing fear UK Muslims might be singled out following this month's call by US Republican presidential candidate Donald Trump to bar all Muslims from entering the United States.
A spokesperson for Cameron said: "We are looking into the issue and the Prime Minister will respond in due course."
The US Customs and Border Protection agency said in a statement issued in London: "The religion, faith, or spiritual beliefs of an international traveller are not determining factors about his/her admissibility into the US,... applicants for admission bear the burden of proof to establish that they are clearly eligible to enter the United States," it added. "In order to demonstrate that they are admissible, the applicant must overcome all grounds of inadmissibility."