In its third-quarter results today the German flag-carrier states that Bishop exercised a put option for his stake of 50%, plus one share, in BMI on 10 October.
This transaction is not expected to complete before 12 January next year, says Lufthansa. Lufthansa already holds just under 30% of BMI and completion will give the German carrier an 80% share.
The agreement on the share option dates back to November 1999.
Lufthansa says the exercising of the option is still subject to regulatory approval and it adds: "Discussions regarding the future structure and options are ongoing between Lufthansa, Sir Michael Bishop and BMI."
Scandinavia's SAS Group holds the other 20% of the UK carrier. The company says it still wants to sell its stake, but has not identified any prospective buyer.
"We are hopeful that Lufthansa and us together can deal with [company] issues more efficiently," says SAS Group head of investor relations Sture Stolen.
"We have expectations that their being in the driver's seat might improve the situation in the company."