These aircraft, which will be used to fly the President, Vice President and Prime Minister, are expected to be in operations from July next year.
At present, President Ram Nath Kovind, Vice President M Venkaiah Naidu and Prime Minister Narendra Modi fly on Air India's 26-year-old Jumbos - B747s. These planes are flown by Air India pilots.
"We are in discussion with the government for providing maintenance services to the VVIP planes. Currently all the modalities are being worked out. We are likely to sign a Memorandum of Understanding (MoU) in this regard in the next one month," the source said.
He said the MoU will be followed by a detailed agreement, which is expected to be in place by March.
"As part of the contract under discussion, Air India will not only be maintaining the planes but will provide components support as well," he said.
The two new B777 planes will come to India from Boeing's US facility, where they are tailor-made for VVIPs, by July 2020.
The B777 planes will have state-of-the-art missile defence systems called Large Aircraft Infrared Countermeasures (LAIRCM) and Self-Protection Suites (SPS).
In February, the US agreed to sell the two defence systems to India at a total cost of USD 190 million.
The two new planes will be flown only by the pilots of Indian Air Force (IAF), an Air India official had said in October.
As many as 4-6 pilots of IAF have already been trained by Air India for this purpose and "some other pilots of IAF will come for training soon," the official had said at that time. IAS RVK