The move follows months of deteriorating relations between the federal government and the Tigray People's Liberation Front (TPLF), the party that runs Tigray.
The two sides fought a war from 2020 to 2022 that resulted in hundreds of thousands of deaths.
On Sunday, the TPLF and six other armed groups across Ethiopia announced they had formed an alliance aimed at overthrowing Abiy's government.
On Wednesday, three local sources, including one security source, told Reuters that Tigrayan forces had taken control of the Mekelle airport from federal police overnight.
The state-owned carrier said on social media that flights to regional capital Mekelle and the cities of Shire and Axum had been halted "due to the current situation in Tigray region", without giving further details.
Reuters could not immediately confirm the situation in Shire and Axum.
Spokespeople for Prime Minister Abiy Ahmed, the federal government and the TPLF did not immediately respond to requests for comment on Wednesday.
The grinding 2020-22 war stemmed from a breakdown in relations between the TPLF, which dominated Ethiopian politics for nearly three decades, and Abiy, whose appointment as prime minister in 2018 ended the TPLF's dominance.
They signed the Pretoria Agreement in November 2022 to end the conflict, but each has been accusing the other of violating the peace deal.
In May, the TPLF seized control of the region's government from the interim administration established under the peace deal.
Abiy's spokeswoman, Billene Seyoum, told Reuters on Monday that members of the newly announced alliance had been collaborating for a while with the financial backing of Ethiopia's neighbour and historic adversary Eritrea.
Eritrea has repeatedly denied accusations it supports Ethiopian armed groups.
