Leo van Wijk, KLM president, told reporters that KLM, and its partners Alitalia and Northwest Airlines, had been invited to submit a bid for a stake in the British short-haul carrier.
The Derby-based airline has said it is also holding talks with the Delta Airlines-Air France grouping and members of the Star Alliance about a link-up to develop its business.
"We're currently in talks with British Midland about how they view their position in an alliance," Mr van Wijk was reported as saying. "They have invited three alliances to come up with a proposition ... We have been asked to make a bid for part of the company."
Scandinavian Airline Service, a member of the Star Alliance, holds a 40 per cent stake in British Midland. Analysts say if British Midland declines to join Star, this stake would pass to the successful grouping.
Industry-watchers play down KLM's chances of success, however, saying that any link with British Midland could face severe scrutiny from regulators over the potential concentration of UK-European routes.
British Midland is attractive to other carriers because it holds about 14 per cent of the take-off and landing slots at Heathrow, the world's busiest airport. A link-up may give fellow alliance members access to British Midland's slots. It could also help British Midland achieve its long-stated desire of running trans-Atlantic services out of Heathrow.
At present, the so-called Bermuda II agreement restricts to just four the numbers of airlines permitted to operate direct services between Heathrow and the United States.
"The first objective is to come up with a joint proposal that would satisfy British Midland," Mr van Wijk was quoted saying. "The intention is to finalise in the next couple of months."
Such a timetable would be slower than British Midland officials have been hinting at. Three weeks ago, the company said an announcement on a deal was "imminent".
Mr van Wijk also said that KLM was on track to merge with Alitalia, the Italian flag carrier. The pair announced an in-depth tie-up in July and operations are expected to start from November.
"We did not commit to this [merger]. But I see it as very likely that we will do it," Mr van Wijk said.
British Midland has been pushing to have Bermuda II re-negotiated, running a campaign of television and newspaper adverts this month.
Some of its 19 code-sharing arrangements, which include a link with Air France, are likely to be scaled back if it joins an alliance.
Code-sharing allows airlines to carry each other's passengers on connecting services and to share computer booking codes.