The Conservatives have been talking to the Liberal Democrats to try to form a government, but Brown said in a dramatic statement delivered in front of his official residence at 10 Downing Street the Lib Dems also wanted to talk to Labour.
The Conservatives, led by David Cameron, won most seats in Parliament but fell short of a majority after an inconclusive election last week.
Labour came second and the smaller Liberal Democrats, led by Nick Clegg, a distant third.
"Mr. Clegg has just informed me that while he intends to continue his dialogue that he has begun with the Conservatives, he now wishes also to take forward formal discussions with the Labour Party," Brown said, adding that he would facilitate those talks.
"I have no desire to stay in my position longer than is needed to ensure the path to economic growth is assured and the process of political reform we have agreed moves forward quickly," Brown said.
"As leader of my party I must accept that that (the election result) is a judgment on me. I therefore intend to ask the Labour Party to set in train the processes needed for its own leadership election," he said.
Brown did not give a precise timeframe for his departure, but said he hoped it would be done by the time of the Labour Party conference, which is scheduled for late September.
Britain's sterling currency fell and government bond futures hit a session low after Brown's comments. Markets had been hoping for a quick deal between the Conservatives and Lib Dems and will not relish the prospect of further delays as parallel talks take place between the Lib Dems and Labour.
Earlier, Liberal Democrat legislators said they were seeking clarification from the party's negotiators about details of a possible deal with the Conservatives.
"Although we are very, very conscious of the need to make these decisions quickly, and that's a clear decision of the parliamentary party today, we also want to make sure that we get these matters right," said Lib Dem legislator David Laws.
Conservative and Lib Dem negotiators said earlier they made progress at talks to reach a power-sharing deal, although others called for caution on how quickly a deal could be clinched.
