"eagerly agree" is correct and usable in written English.
You can use it to express enthusiasm for an agreement. For example: "I eagerly agree to your proposal."
Ms. Felsen, by then recovered, eagerly agreed.
When it came time for dessert, Mr. Colavito and Ms. Favors eagerly agreed to share tiramisù.
Mr. Wilk eagerly agreed to coordinate the program, which started at the Greens Farms Academy concert hall.
When Torre asked for a guaranteed third year this week, the Yankees eagerly agreed.
Noreen Naroo-Pucci, who helped put the collection together, eagerly agreed.
When Mayor Richard Riordan, a good friend of Broad's, asked him to take over the project, he eagerly agreed.
When he asked Ms. Levis to participate, she eagerly agreed and quickly began rounding up the old gang of Moguls.
For the sake of continuity, Colletti said, he wanted the successor to come from Torre's staff, and he eagerly agreed to let Torre bring Mattingly.
At one point during his first visit, Frank condemns "the hopeless emptiness of everything in this country," and John Givings eagerly agrees: "Now you've said it.
