The phrase 'genuine eagerness' is correct and can be used in written English.
It means a sincere and enthusiastic desire or willingness to do something. Example: John showed genuine eagerness to learn the new software, spending hours practicing and asking questions to improve his skills.
When I made a coaching visit to Argentina in March, it was immediately obvious there was a genuine eagerness to think outside the box.
You can count on your fingers the writers whose work is so rare, in both senses of the term, that reviewers greet with genuine eagerness the announcement of a new book from their pens.
As with all other luxury hotels in Moscow, the staff at the Kempinski is multilingual and performs all duties, from delivering your dry cleaning to instructing you how to negotiate with a cab driver, in a way that conveys a genuine eagerness to help.
His command of the issues, coupled with his tireless devotion to constituent concerns, underscores a genuine devotion to and eagerness to fight for his native city.
What kept her from being a poule de luxe pure and simple, Powell thought, was an odd streak of seriousness, a half-buried intellectual twist that allowed her to combine a relish for causing trouble for its own sake with a genuine shyness, uncertainty and eagerness to learn.
From his initial eagerness to remove the genuine social evils of slum housing, Poulson had grown, step by step, into a full-scale criminal.
Living away from home against her parents' wishes, engaged to a white non-Muslim, she struggles with how to tell her parents about him and their plans, a problem that symbolizes the conflict between her Muslim background and traditional family and her eagerness to be a genuine American.
Its ideas are unified by the author's genuine fascination with his material and his eagerness to follow it wherever it leads, even when it goes half-mad.
And Syd Brisbane's Manuel captures his eagerness and unfortunate ineptitude with genuine warmth – although all the violence and disparagement he receives really isn't funny any more.
"The key qualities you want from supporters are loyalty, an understanding of the position the club is in, a genuine will to win and a ready eagerness to provide whatever support they can to help their team.
Thus, Mr. Kim's seeming eagerness to return to talks, if genuine, could disrupt Seoul's plans.
