A total of six Bucks players were isolating on Thursday meaning the small squad was at breaking point ahead of Saturday's game with Southport.
The club spoke to National League officials to see if both games could be called off and confirmation was received on Friday morning.
In a statement, the club said: "AFC Telford United can confirm tomorrow’s game against Southport and Tuesday’s night’s game against Curzon Ashton have been postponed due to COVID-19 within the Bucks camp.
"Tickets that have been purchased will be valid for the rearranged game which will be announced in due course."
It means the club won't play at home until January 22 when they face Darlington.