A PR agency source has suggested that the agency has been selected but negotiations are continuing.
Fleishman-Hillard will be ending the current £1.5m contract this month, but the fees are expected to be reduced.
The four territories of Europe, Asia Pacific, Africa and the Middle East could be covered by between one and four agencies.
Currently BA does not use an agency in the UK. BA US currently works with Agnes Huff Communications Group on the West Coast, as well as Weber Shandwick on crisis.
BA uses Fleishman-Hillard in all other international regions.
BA admitted that discussions were taking place with the agency.