He was desperately anxious to learn, and had little faith in his own intuition.

They were all like that, desperately anxious to win favor.

He was desperately anxious for Viola to fall in love with him.

I didn't know what they were talking about, but I was desperately anxious not to be left out of anything.

He was desperately anxious to get back to Division headquarters and gain control of the battle.

Being desperately anxious to get the thing over one way or another, at the proper moment I took the first of them.

I know only too well how desperately anxious you must be.

He was desperately anxious to learn whether I had understood those whispered words.

They've got something they're desperately anxious I shan't find out.

I would say that both entities fully understand the situation and are desperately anxious to cooperate.