They say that representatives of Mr Mugabe, Zimbabwe's military chiefs and the opposition have held meetings chaired by South Africa's president.
The sources say Mr Mugabe would give an address to the nation but urge caution until the announcement has been made.
The opposition says it won Saturday's general elections.
Under the proposed deal, Movement for Democratic Change leader Morgan Tsvangirai would be declared the winner of the presidential race after Mr Mugabe had stepped down.
President Mugabe, 84, came to power 28 years ago at independence, but the economy has been in freefall in recent years.
The Zimbabwe Electoral Commission has not yet given the results of the presidential race, sparking MDC claims that the outcome was being fixed.
BBC correspondent Ian Pannell has spoken to three MDC sources who have confirmed that a deal had almost been reached.
One was 90% sure that this would happen but others were less confident.
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