Jamieson, who lost his place to on-loan Rangers 20-year-old Liam Kelly, had been sent out on loan to junior outfit Bo'ness United earlier this month.
He would become Accies' third goalkeeping addition of the summer.
Gary Woods was signed permanently from Leyton Orient and Remi Matthews on loan from Norwich City.
The Premiership club lost Michael McGovern to Norwich this summer after his Euro 2016 displays for Northern Ireland.
Meanwhile, 16-year-old Jon McCracken followed him from New Douglas Park to Carrow Road earlier this month.
Jamieson had made 101 appearances for Livingston, 16 while they were in the Championship last season.

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Large house parties could also be vulnerable to sneak-in thefts over the festival period.
The warnings are in a safety message from police in the Highlands and Islands.
Police Scotland said criminals targeted homes at this time of year to try and steal presents and money.
Insp Ewan Henderson said posts and other messages on social media sites before and during parties could alert thieves.
He said: "Christmas nights out are planned in advance, so just taking a few minutes to think about the security of your home will mean everyone's Christmas can be fun-filled.
"Posting photos on social media of nights out are an everyday occurrence.
"What we are asking is that you think about it before you post the messages. Are you inviting someone to break in to your home by advertising it is un-occupied? Could the messages be posted the next day?"
Peter Kirwan, of Neighbourhood Watch Scotland, added: "We're encouraging people to ask a neighbour to keep an eye on their house when they go away.
"Small things like moving curtains and taking in mail can help make a place look occupied and reduce the risk of a house breaking."
Goalkeeper Darren Jamieson has been released by Livingston, with the League One club saying that the 25-year-old is to join Hamilton Academical.
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Parts of Mexico City's historic centre have been closed to traffic as film crews shoot scenes for a new Godzilla film in the Mexican capital.
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Christmas party selfies and other social media posts about festive nights out could alert thieves to unoccupied homes, police have warned.