Black Cats chief executive Martin Bain revealed in an interview this week that funds from the £30m sale of Jordan Pickford over the summer have been entirely used to finance the day-to-day running of the club following its drop into the Championship.
American owner Ellis Short has put the side up for sale and has reportedly refused to pledge additional injections of cash but Grayson, who replaced David Moyes in the dugout in June, says that the situation was made clear to him from the outset.
"I came to this football club because I wanted to, and Sunderland chose me as the right man to come here," he told reporters. "I knew the parameters we were going to be working in so it has been no great surprise. We've had to change the squad around, and that's what you have to do.
"Look at Hull City for example. They had to make a lot of changes on and off the pitch as well. It comes with the territory of relegation from the Premier League.
"Martin has put it out there what happened in the past. What we have to do now is focus on the future and make sure everybody is heading in the right direction, in a positive manner, and results will help us do that."
Sunderland's return to the second flight of English football has failed to get off to the best start, however, with the club just outside of the relegation zone after eight games, but Grayson insists that he is still the right man for the job.
"I feel I have a group of players that can still have a successful season. It's probably more turning the performances of nearly good results into very good results. That's what I'm here to do," he added.
"I like to think I'm a good coach, a good manager, and that's the reason why I'm here, to make the team better. I feel that over the next few weeks and months we will do that."
Next up for Sunderland is a trip to Everton in the EFL Cup tomorrow night, ahead of a home encounter with high-flying Cardiff City in the league on Saturday.