There have been a range of problems at its main installations including Barts and the London NHS Trust, Royal Free Hampstead NHS Trust, Barnet and Chase Farm Hospitals NHS Trust and Queen Mary's Sidcup.
NHS London said that, since BT is already stretched to deliver, it "would be unhappy if BT tried to take the commercial advantage following the cancellation by Fujitsu of their contract".
But BT is negotiating with health officials to take over the support of eight trusts in the south of England where Fujitsu was the main supplier. It could win further business at NHS trusts where Fujitsu was due to be the supplier.
The claim that BT is stretched to deliver was made in a paper to the board of the Royal Free Hampstead NHS Trust. The Royal Free was the first NHS site in England to go live with the first release of the Cerner Millennium Care Records Service in June. The "LC1" version of Millennium has been tailored for use in London.
BT denies the claim. Its spokeswoman said: "The BT answer to "are we stretched" is 'no'. We're confident we have the right resources in place to deliver our commitments in London. If we acquire additional responsibilities in the south we will make sure these are properly underpinned and fully resourced while continuing to develop, deploy and manage all our systems in London to the required standard."