Cameron told the BBC that if Begg had information about "who these people are, he should provide it."
Begg said he will help the government secure the release of British hostage Alan Henning from IS, but was prevented from issuing a direct appeal. Henning was later beheaded on camera by his captors in a video disseminated last Friday.
The former Guantanamo inmate said the British government had refused to deliver a public message to IS in an attempt to save Henning. Begg said British foreign office asked him to convey a private message to the militant group through an intermediary, but that he refused on the grounds it would not have the desired effect.
Cameron: Govt happy to work with anybody to release hostages
The prime minister said that while his government was "very happy to work with anybody" to secure the release of hostages, "as far as I know, most of Begg's messages delivered to IS were ignored."
Begg said he was approached by Henning's friends in December 2013, just after the 47-year-old taxi driver was taken hostage. The former Guantanamo detainee said he believed he had an idea about who kidnapped Henning.
Meanwhile, Henning's family has slammed the British government for preventing them from speaking publicly about his capture in recent months.
When asked about the government's policy on hostages, Cameron said: "I know as prime minister that we did everything we could to try and find him and to try and get him, but we weren't successful."
He said the government's approach had been to work with families and not raise the media profile to avoid putting hostages at "even greater risk."