Telecoms company BT is running the UK-wide contest to mark the 80th anniversary of the telephone service, which receives 12 million calls a year.
The winner will be only the fifth person to give the time "at the third stroke" when people dial 123.
Sara Mendes da Costa has been the voice of the speaking clock since winning the last competition in 2007.
Previous voices have included an actor and a London telephone exchange supervisor.
Listen to the current speaking clock
At the third stroke... Why do people still dial for the time?
The speaking clock service provides the precise time, announced every 10 seconds at any time of day or night.
It began on 24 July 1936 as a service for people who did not have a watch or clock to hand.
Before 1936, people used to ring the exchange operator - a real person - to settle a dispute over the time.
Now officially called Timeline, the speaking clock has never been a free service - in the early days, calls were one penny from home and tuppence from a phone box.
It now costs 45p a minute from a BT landline, and still experiences peak periods on Remembrance Day, New Year's Eve and when the clocks go forwards or back.
Ms Mendes da Costa, who will be one of the competition's judges, said she was sad her time as the voice of the "iconic service" was coming to an end, but "10 years is not a bad run".
"And it'll be great to be on the judging panel to choose the new voice, just as my predecessor Brian Cobby helped choose me," she said.
The speaking clock gets its time from the atomic clocks at the National Physical Laboratory, the UK's official time-keeper.
To celebrate the 80th anniversary of the service, BT is donating the original speaking clock machines from 1936 and 1963 to the British Horological Institute.
The competition, which is being run in partnership with the BBC's The One Show in aid of Children in Need 2016, is open to anyone aged 10 or over. Details of how to apply are available on the BBC One Show website.
The competition closes at 22:00 BST on Monday 29 August 2016, with the winner to be announced in November.
A competition to become the new voice of the speaking clock has been launched.