Hammond's remarks came as Britain prepares to host a visit by Egyptian President Abdel Fattah al-Sisi this week.
Egypt, a close ally of the United States and the most populous Arab country, dismissed a similar claim of responsibility for the crash by Islamic State on Saturday.
"It is believed to be an explosion but what kind is not clear. There is an examination of the sand at the crash site to try and determine if it was a bomb," said an Egyptian source who is close to the team investigating the black boxes.
"There are forensic investigations underway at the crash site. That will help determine the cause, to see if traces of explosives are found."
Sisi has described Islamist militancy as an existential threat to the Arab world and the West and has repeatedly called for greater international efforts to combat the militants.
Hammond said Britain is "advising against all but essential travel by air through Sharm el-Sheikh airport. That means that there will be no UK passenger flights out to Sharm el-Sheikh from now."
Remarks earlier on Wednesday by Britain's Cameron, who was due to hold talks in London with Sisi on Thursday, of concerns "the plane may well have been brought down by an explosive device" drew criticism from Egyptian Foreign Minister Sameh Shoukry.
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He told CNN he was "somewhat surprised" by the British statement.
"This is a matter for the investigation to clarify and we should not prejudge or take any measures that might have implications," Shoukry said.
"Implication also that the fact that a very large number of Egyptians who rely heavily on the tourist industry."
Britain said it was working with airlines and Egyptian authorities to put in place additional security and screening measures to allow Britons in Sharm el-Sheikh to get home, but that would take time and there would be no flights returning from the resort on Thursday.
A Russian aviation official said the investigation was looking into the possibility of an object stowed on board causing the disaster.
"There are two versions now under consideration: something stowed inside (the plane) and a technical fault. But the airplane could not just break apart in the air - there should be some action. A rocket is unlikely as there are no signs of that," the Russian official said.
Security experts and investigators have said the plane is unlikely to have been struck from the outside and Sinai-based militants are not believed to possess the technology to shoot down a jet from a cruising altitude above 30,000 feet.
Any evidence that a bomb knocked the plane out of the sky would deal a heavy blow to tourism in Egypt, a pillar of the economy that is struggling to recover after years of political turmoil, and would also undermine Sisi's assertions that Cairo has brought under control Sinai Province's insurgency.
Sinai Province has killed hundreds of Egyptian soldiers and police since Sisi, as army chief, toppled Islamist President Mohamed Mursi in 2013 after mass protests against his rule.
Sisi was elected president last year on promises he would stabilize Egypt and rebuild its shattered economy. Critics say his tough crackdown on Islamists will only create more radicals in Egypt, which has fought militants for decades.
Investigators have extracted and validated the contents of the flight data recorder, one of two so-called black boxes recovered from the Russian plane, Egypt's Civil Aviation Ministry said.
Read MoreISIS Bomb May Have Downed Russian Jet, U.S. Official Says
The ministry said the second black box, which contains the cockpit voice recorder, was partially damaged and much work was required to extract data from it.
Sinai Province has said it had brought down the airliner "in response to Russian air strikes that killed hundreds of Muslims on Syrian land."
In an audio message posted on a Twitter account used by the group, Islamic State's Egyptian affiliate insisted on Wednesday it was behind the crash. The claim could not immediately be authenticated.
"We, with God's grace, are the ones who brought it down, and we are not obliged to disclose the mechanism of its demise," the speaker said.
Russia, an ally of Syrian President Bashar al-Assad, launched air raids against opposition groups in Syria including Islamic State on Sept. 30. The hardline group has called for war against both Russia and the United States in response to their air strikes in Syria.
Late on Wednesday the Irish Aviation Authority (IAA) directed all Irish airlines on Wednesday not to fly to or from the Sinai Peninsula until further notice. The Russian-operated plane was registered in Ireland and the IAA is taking part in the official investigation into the crash.