Cox was brutally attacked by Roma fans before the Champions League semi-final at Anfield last season, and has been in hospital since then. Coleman gave €5,000 to an online fund for the Irishman.
"You think 'Do you put your name towards it or not?' You might get people saying 'he might have put more money in' or whatever," Coleman said.
"I wanted to put my name to it because sometimes that raises more publicity. That's what the Liverpool manager did.
"We have to be together. I think football is great for sticking together.
"I've had people in the streets, Liverpool fans, stop me and say 'thanks very much'."