Dunmore was chief executive of the New Opportunities Fund until it merged with the Communities Fund to form the BLF in 2003.
Starting his career in housing and urban policy, Dunmore moved to a post at the Merseyside Task Force and was later appointed director of regeneration, transport and planning in the Government Office for Merseyside.
In a statement to potential applicants to replace Dunmore, BLF chairman Sir Clive Booth said: "As chief executive, you will want to see that our funding activities continue to combine dynamism and innovation with probity and rigour - and that we listen, learn, and share best practice with our partners and stakeholders."
"You will want to see also that our processes and customer care are second to none and that ours is a modern, responsive organisation," Booth added.
The advertisement describes the vacancy - which has a salary of £150,000 as well as a performance bonus as "the opportunity of a lifetime".
A BLF spokeswoman said that the Fund hopes to find a successor by November.
More information on the post can be found here.