The new 112-bed wing, which will cost 700,000 pounds ($973,000) and be compliant with United Nations standards, will make it easier for Britain to comply with a prisoner transfer agreement it signed with Nigeria in 2014.
According to Ministry of Justice figures, there were 320 Nigerian prisoners in British jails at the end of 2016, accounting for 3% of the foreign prisoner population. Kirikiri is not one of the oldest prisons in Nigeria but it does date back to colonial times.
The UK has made previous deals with some countries to transfer prisoners in UK jails to their countries of origin.
A Foreign Office spokesperson said: "The government believes that wherever possible foreign nationals should serve their sentences in their own country".
Reuters reported Thursday that in a written statement to parliament, British Foreign Secretary Boris Johnson said tenders had been placed and a supplier identified to conduct the building work at Kirikiri in Lagos, but the supplier remained unnamed.
In support of this, and as part of efforts to improve the capacity of the Nigerian Prison Service, the United Kingdom government has come up with plans to build a prison in Nigeria to enable it to transfer prisoners from Britain.
In 2015 the Government pitched an ambitious £25m project to build a 1,500 bed prison in Jamaica, with the aim of sending more than 300 Jamaican inmates with sentences of more than four years there.
Nigeria's Kiri Kiri prison was built in 1955 and has a long history of overcrowding.
"The provision of this assistance is in line with the Government's security and stability objectives in West Africa".
The new wing is being paid for using the CSSF (Conflict, Stability and Security Fund) migration returns fund, said Mr Johnson.