After their 2-2 draw with Leicester City, owner Evangelos Marinakis was seen confronting manager Nuno on the pitch.
It was later revealed Marinakis had concerns about the medical staff and their handling of injured striker Taiwo Awoniyi. The Nigerian was taken to hospital on Monday for emergency abdominal surgery.
Clough, a former Forest striker and son of managerial legend Brian Clough, told talkSPORT: "It's not (a good look) when it's done in public.
"Everything like that should be handled behind closed doors. Then it's much easier for everybody to take.
"To see the owner on the pitch isn't ideal. It shows the sort of pressure that comes with this Champions League qualification.
"£75million comes along with it, that's a few quid worth to the owner and to the club, so you understand it."
Cleared up?
Clough, now in charge of Mansfield Town, also said: "I'm glad they cleared it up straight away, it was frustration over the substitution, the injuries and everything.
"You have to rely on certain things, your medical staff, the player himself and then make decisions accordingly.
"I think more of the blame is on them than anybody."
Clough continued: "But don't let this lose sight of the incredible season Forest have had - if you'd have said nine months ago that you'd be in an FA Cup semi-final and in with a chance of Champions League, you'd have said absolutely no chance.
"Never lose sight of the progress they've made."