LONDON—The British government will move toward participating in Syrian airstrikes if there is a political consensus to do so, a senior minister said Sunday, the latest sign of building momentum within the U.K. to step up military action.

George Osborne, U.K. Treasury chief and close political ally of Prime Minister David Cameron, said the West hadn’t done enough to contain Islamic State militants who control territory in Syria and Iraq. Speaking in an interview with the British Broadcasting Corporation, Mr. Osborne said Mr....