The deal, which requires regulatory approval, would give IAG more than half the daily take-off and landing slots at Heathrow, Europe's busiest airport and which is operating at full capacity after plans to build a third runway were scrapped.
Virgin Atlantic, which tabled a competing offer earlier this month, said it would ask anti-trust regulators to block the deal as it would stifle competition and push up prices.
"We will fight this monopoly every step of the way as we think it is bad for the consumer, bad for the industry, and bad for Britain," Virgin's Richard Branson said.
IAG currently holds 43.1 percent of the slots at Heathrow and would have around 52 percent of the hub's slots if its bid for bmi gets approval from regulators.
The group, which also owns Spanish carrier Iberia, said absorbing bmi's runway slots would give it the option of launching lucrative new long haul routes from Heathrow, and that it aimed to complete the deal by March.
Willie Walsh, IAG Chief Executive, said there would be "some job losses" as part of an effort to turn around ailing bmi, which last year racked up pretax losses of 153 million pounds.
IAG shares were up 0.96 percent at 146.5 pence at 1030 GMT, in line with the FTSE Eurofirst 300 index of top European shares. Lufthansa was 1.9 percent higher.
"I think this is a highly positive strategic move for IAG because it delivers a big boost to its presence in what is arguably the world's most attractive aviation market," said Douglas McNeill, an analyst at stockbroker Charles Stanley in London.
"I think Lufthansa will regard the price as quite a good outcome for them as well because they've managed to get a cash price for something that is heavily loss-making."
Lufthansa has the option of selling bmi's 'bmi regional' and 'bmibaby' units before the deal completes, and IAG will pay a lower price if those business are not offloaded, IAG said
Under the deal, the German airline also holds onto bmi's staff pension liabilities.
