The deal is expected to be signed by tomorrow morning.
Murphy founded the agency in 2011 together with former Number 10 colleagues and focused its offering on government, corporate and NGO communications.
The acquisition dramatically strengthens Bell Pottinger’s connection with Labour politics as the party maintains a narrow lead in the polls with 11 months to go until the General Election.
Murphy spent three years in Number 10 under Blair, while Centreground boasts several Labour figures including former Home Secretary Jacqui Smith and former Blair chief of staff Jonathan Powell.
Bell Pottinger chief executive James Henderson declined to comment on the deal.