The waste company has already recommended a £1.2 billion offer from Montagu and Global Infrastructure Partners.
Terra Firma and Suez had been planning on offering a counter-bid offer for Biffa but after undertaking due diligence have shelved plans to take over.
In a statement on the acquisition update, Biffa said: The Biffa Board has now been informed by those third parties that they are no longer working towards making a competing offer for Biffa. As a consequence, the Biffa Board is now not aware of any third parties who are working towards making such a competing offer.
This now leaves Montagu and GIP as the frontrunners in the battle for Biffa. The Montagu and GIP proposal will now proceed to a court hearing, held by Biffa shareholders, due to be held on March 12 2008. According to a Biffa spokesman, the outcome of the proposed takeover will be decided during the middle of April.