A video shared on social media and verified by Reuters showed the planes on the airport tarmac.
One of the diplomats said that Somalia was "playing with fire" by importing the Egyptian arms and antagonising Ethiopia.
Somalia and Egypt's foreign ministries did not immediately respond to requests for comment.
Egypt's offer to contribute troops to a new peacekeeping mission to be launched next year in Somalia was announced in an African Union communique earlier this month. Cairo has not commented on the matter publicly.
"If the Egyptians put boots on the ground and deploy troops along the border with Ethiopia, it could bring the two into direct confrontation," said Rashid Abdi, an analyst with the Sahan Research think-tank.
"The threat of a direct shooting war is low, but a proxy conflict is possible."
Ethiopia's foreign affairs ministry said in a statement that the country "cannot stand idle while other actors are taking measures to destabilise the region", saying it had worked to promote peace and security for Somalia and the region, including discussions to resolve differences with Somalia.
"Instead of pursuing these efforts for peace, the Government of Somalia is colluding with external actors aiming to destabilise the region," the ministry said.
Ethiopia's statement did not mention Egypt or its arms delivery to Somalia.
Turkey has hosted two rounds of
indirect talks since July between Somalia and Ethiopia over the Somaliland deal, which is yet to be finalised. A third round is expected next month.
Landlocked Ethiopia says it needs access to the sea. Mogadishu insists that Somaliland, which has not obtained international recognition despite enjoying practical autonomy for more than 30 years, is part of Somalia.
Source: Reuters