LONDON — Shaken by polls showing momentum shifting toward independence for Scotland, the British government will offer proposals for greater political and fiscal autonomy for the Scots if they vote to remain within the United Kingdom in a referendum on Sept. 18, George Osborne, the chancellor of the Exchequer, said on Sunday.

The narrowing polls have caused considerable anxiety among politicians and business leaders, driving down the value of the pound and raising questions among investors about the stability of the economy and the fate of the current British government.

The vote, which could bring an end to the 307-year union between Scotland and England, is also regarded as important to the future of the British prime minister, David Cameron. As leader of what is still formally known as the Conservative and Unionist Party, Mr. Cameron, already facing internal divisions over Britain’s membership in the European Union, may not survive politically if Scotland votes to break away from the United Kingdom in a referendum that he negotiated with the Scottish National Party and its leader, Alex Salmond.

On Sunday, Mr. Osborne, a close ally of Mr. Cameron’s, responded to the tightening race by promising more powers to Scotland if it votes no.