Byrne has been in charge at the County Ground since Maurice Malpas was relieved of his duties the week before last, since when the Robins have featured in successive 2-2 draws with Leicester City and Bristol Rovers.
"The manager's job is the natural progression for me after my coaching career so far and I hope I can give the chairman something to think about," he said.
"I expect to be in charge for our next game on Tuesday night, but after that I have no idea what the future holds."