Mr Devine has been barred by party bosses from standing again as a Labour candidate after allegations over his expenses.
Now the local party is due to choose a new candidate before the end of the month.
Cllr Murray announced he was throwing his hat in the ring after Labour decided against ordering an all-women shortlist.
Former West Lothian council leader Graeme Morrice has already said he will seek the nomination. Other names mentioned as possible contenders include Kevin Lindsay, Scottish officer for rail union ASLEF and West Lothian councillors John McGinty and Cathy Muldoon.
Cllr Murray, first elected to the city council representing Alnwickhill ward in 2003 and now councillor for Liberton/Gilmerton, said he believed in the wake of the expenses scandal people wanted to see change in politics.
He said: "There is an appetite for politicians to have wider experience of the world, business and so on. We should be bringing through fresh, young blood to reinvigorate not just the Labour Party but politics as a whole."
Cllr Murray runs his own events management and web-casting company 100mph Events and also the Aspen cafe bar in South Bridge.
Would-be candidates have until Friday to make their interest known and the selection will be made by the end of the month.
Labour's selection process normally takes 12 weeks but when an election is due within six months, the process is cut to four weeks.
Labour bosses have recently approved all-women shortlists in Airdrie & Shotts, where former Home Secretary John Reid is standing down, and Kilmarnock & Loudon, where former Scottish Secretary Des Browne is quitting and Labour MSP Cathy Jamieson is hoping to win the nomination.
But the party decided against any restriction on the Livingston selection.
Labour said of the last six seats to come up for selection, an all-women selection had been ordered in three and an open selection had been allowed in the other three.
Mr Devine had a majority of 2,680 when he won the 2005 by-election following the death of Robin Cook. But the SNP won Livingston at the Holyrood elections in 2007.
Labour believes it can hold on to the seat despite Mr Devine's expenses.
A party spokesman said: "There is a very strong Labour vote in Livingston. At the last general election Labour had a very significant majority and we would expect to hold that.
"Labour took very tough action and we are providing Livingston with a new candidate because we expect the highest possible standards from our MPs."