Kindeya Geberehiwot, a member of the TPLF’s central council, told BBC Tigrinya that the siege happened in two parts of the city on Monday morning.
According to Mr Kindeya, three people were killed and many were injured in the first hit, which occurred on the outskirts of Mekelle at approximately 09:30 local time (06:30 GMT).
He said that the second attack hit a crowded public market, injuring a number of people.
Occupants have also alerted the BBC about the attacks, and online media users have provided images purportedly illustrating the damage inflicted.
Tigray’s cell phone network is down, making it difficult to easily confirm cases.