It is one of the most powerful roles in British business and comes just months after Mr Burrows issued a grovelling apology to shareholders of the bank, who have seen the bank reduced to a shell.
Mr Burrows is tipped to replace Jan du Plessis, who has taken over as chairman of the Rio Tinto mining group.
Mr Burrows resigned from Bank of Ireland at its annual meeting in May to be replaced by former chief executive Pat Molloy, having first offered a humble apology to angry shareholders for falling profits and share prices.
The appointment of former bankers to top positions in Britain has not gone smoothly. The decision by Alliance Boots to hire Andy Hornby, who led HBOS into its forced merger with Lloyds TSB, was widely criticised.
According to the Financial Times, BAT’s biggest investors are understood to have accepted Mr Burrows as someone with a good boardroom pedigree and his appointment is expected to be confirmed before the weekend.
Mr Burrows is a former chief executive of French drinks giant Pernod Ricard and is a board member of Carlsberg and Rentokil Initial, the pest control group. For 22 years he was chief executive of Irish Distillers, where he pioneered the globalisation of the Jameson Irish whisky brand. He also oversaw the successful sale of the group to the French group, Pernod Ricard, following a hostile bid from a number of British drinks firms.