I'm not an authority on this, and it would depend on the NHS Clinical Commissioning Group approving it, but from what I recall it would be possible to have it at home, yes.
Incidentally, however, if Social Services kindly suggest to you that you might like to care for your relative at home, rather than the care home they are at, you should take caution. We got offered that twice, and the first time we fell for it, made all the plans etc grateful to take the opportunity while it was on the table. Turned out it was a ruse - they do this because it makes it easier to obtain a court order stopping you from moving your relative. They can do that if they can insinuate that (or just lie) and say they suspect you want to move your aged relative back home, making out you might abscond with them. They can't do that so easily if you are simply asking to move your relative from one failing care home to another, as they don't have that excuse.
So don't be surprised if you have a meeting with the local Safeguarding team and they suggest you move your rellie back home. We got that a second time, even when we obviously were looking to only move our mother to another care home. When we looked at the minutes of the meeting, we found that, again, we falsely quoted as claiming to want our mother back home when we had never asked for any such thing. There were other little lies too, suggesting our mother was in a very frail state, all aimed at building an argument stopping us from moving her to another care home.
I do suspect that the way CHC is run in care homes, there is a good chance they won't have to pay out for very long. However, cared for at home, with loving relatives keeping an eye on the situation, and they could well be out of pocket.
Less cynically, home care does not have a great reputation anyway, as you hear of visits lasting only 10-15 minutes, so care at home may not be practical.