The Dublin-Heathrow route is one of the busiest airline routes in Europe and the slots are currently worth an estimated €400m.
But as part of a proposal to convince the European Commission to allow Ryanair take over Aer Lingus, it is offering to sell 20 of Aer Lingus's 24 landing slots at the UK's busiest airport to British Airways.
The EU is concerned that if Ryanair were to take over Aer Lingus, the new entity would have a monopoly on flights to the UK. This latest proposal is aimed at alleviating this fear.
However, even if the EU does approve the takeover, Ryanair still faces a huge battle to persuade Aer Lingus shareholders and staff (who are also shareholders) that it is in their best interests.
If the takeover were to go ahead, Aer Lingus would be left with only four slots at Heathrow.
Aer Lingus may decide to use them for alternative services rather than for Dublin- Heathrow routes. This would mean that for the first time in half-a-century, the Irish flag carrier would no longer operate services between the two airports.
Ryanair, which owns 29.8pc of Aer Lingus, is trying to secure regulatory approval for its third takeover bid.
The European Commission last month objected to Ryanair's bid on competition grounds. It prohibited Ryanair's first bid in 2007, and Brussels has never cleared a merger that it previously rejected.
The proposed deal between Ryanair and International Airlines Group (IAG), the parent company of British Airways, would see BA take over Aer Lingus services from Dublin, Cork and Shannon and run them as normal for a period of between three and five years.
With BA operating these services, Ryanair believes that a combined Ryanair-Aer Lingus would not be dominant on those routes, thereby alleviating any EU competition fears.
The dramatic overhaul of routes between Ireland and Britain would see as many as six BA planes and four from Flybe based in the Republic.
Heathrow is running at near capacity, so the opportunity to buy Aer Lingus's slots there could provide BA with an important means to eventually expand its long-haul services. Aer Lingus is the third-biggest airline at Heathrow.
BA is offering to take responsibility for many of Aer Lingus's services out of Heathrow for at least three years.
After that it would have the right to buy the airline's Heathrow slots and re-allocate them to different destinations, such as New York.
A separate part of the same takeover process being planned would see Flybe operate flights to and from Ireland on 20 other routes where Ryanair and Aer Lingus currently both have services for at least three years.
These routes are also from Dublin, Cork and Shannon. This move would also remove the monopoly any merged Ryanair-Aer Lingus operation would have on those routes.
The moves by BA and Flybe are understood to form the centrepiece of a remedies package that Ryanair has submitted to the European Commission to try to persuade Brussels to approve the takeover bid.
A spokeswoman for IAG told the Irish Independent: "We have signed a non-binding memorandum of understanding with Ryanair which is subject to EC approval, as part of its review of Ryanair's proposed takeover of Aer Lingus.
"The memorandum of understanding is also subject to IAG board approval."
Ryanair declined to comment.