If faith is a gift from God, and atheists (might) go to hell, is God ultimately willing to send atheists to hell? (Catholic perspective) Premise 1: Faith is a gift from God. So says St. Thomas Aquinas here. Also, according to the Dogmatic Constitution of the Catholic Church De Fide Catholica Et De Ecclesia Christi:

... the Catholic Church teaches that this faith, which is the beginning of man's salvation, is a supernatural virtue, whereby, inspired and assisted by the grace of God ...
Wherefore, faith itself, even when it does not work by charity, is in itself a gift of God...

Premise 2: Atheists (who have heard of Jesus) (might) go to hell. Related questions here and here.
Conclusion: Atheists (who by definition have no faith in Jesus and in its Body, the Church) might go to hell because they have not been inspired and assisted by the grace of God. i.e. they have not received faith.

Thus, if faith is a gift from God (i.e. it cannot be acquired through natural means but only through supernatural intervention), according to Catholic belief, are Atheists (and Agnostics and any non-Christian) not to be blamed for their lack of faith? This is, if it is not their fault, why should they, in principle, go to Hell?
By choosing to turn one's back on God, one condemns oneself ...


  CCC 1033 To die in mortal sin without repenting and accepting God's merciful love means remaining separated from him forever by our own free choice. This state of definitive self-exclusion from communion with God and the blessed is called "hell." 


An atheist does not repent (repentance ~ to turn toward God, or to return to God), nor accept God's love (how can one accept love from a person whom you insist does not exist?) and therefore remains separated from God forever by their own free choice.  Unlike the Calvinists, the Catholic Church believes in free will.  The willfulness figures into the self-condemnation, and so the asserted blamelessness in the question has no standing in Catholic belief.  


Catholics do not subscribe to "once saved always saved." Even with the gift of faith, from God, and the remission of all sins at baptism the Catholic Church teaches that we are all sinners and that we can screw it up. (CCC 1704 - 17091).  The sacrament of penance and reconciliation has a purpose: to restore the state of being in communion with God.  The question has an embedded assumption of a lack of agency on the part of the atheist.  That is a Determinist assumption that is not valid for Catholic belief which embodies free will.    


While there is more to this regarding particular judgment, which I covered here, in a nutshell God doesn't condemn the atheist: the atheist condemns himself, or herself.  To use a bit of imagery: the hand of God is there, extended, in the eternal offer of friendship and communion.   

An atheist is not ignorant of God. (An agnostic could make that claim).  An atheist rejects God.  In theological terms, that is a willful turning away when one considers Catholic teaching.  The offer remains there, even with the face and heart turned away from God.  

This is God's will (as stated in the CCC):  


  CCC (Prologue) 2: God our Saviour desires all men to be saved and to come to the knowledge of the truth. (ref 1 Tim 2:3-4.)


The grace that God offers aligns with the point in your question.  I want you to come to me is God's offer.  The atheist (or anyone for that matter) may accept or not accept that offer. The acceptance is an exercise of free will in Catholic belief.     

From the opening statement of belief in the CCC:  


  I. The life of man - to know and love God
  CCC 1 God, infinitely perfect and blessed in himself, in a plan of
  sheer goodness freely created man to make him share in his own blessed
  life. For this reason, at every time and in every place, God draws
  close to man. He calls man to seek him, to know him, to love him with
  all his strength.  He calls together all men, scattered and divided by
  sin, into the unity of his family, the Church. To accomplish this,
  when the fullness of time had come, God sent his Son as Redeemer and
  Saviour. In his Son and through him, he invites men to become, in the
  Holy Spirit, his adopted children and thus heirs of his blessed life.


The atheist rejects this call, and this gift, freely.  


  CCC 1022 Each man receives his eternal retribution in his immortal soul at the very moment of his death, in a particular judgment that refers his life to Christ: either entrance into the blessedness of heaven-through a purification or immediately, -- or immediate and everlasting damnation. 


This was discussed in some detail in this answer here.  

What is Grace?  

Grace is an undeserved gift from God.  


  CCC 1996 
  Our justification comes from the grace of God. Grace is favor, the free and undeserved help that God gives us to respond to his call to become children of God, adoptive sons, partakers of the divine nature and of eternal life. [cf. John 1:12-18; 17:3; Romans 8:14-17; 2 Peter 1:3-4.]  


You asked about faith as being gifted, assisted, even inspired by Grace.  If one rejects the gift, it is by an action of the will: as above, we are back to free will.  



1 Some teaching on Free Will ... 


  CCC 1704 The human person participates in the light and power of the
  divine Spirit. By his reason, he is capable of understanding the order
  of things established by the Creator. By free will, he is capable of
  directing himself toward his true good. He finds his perfection "in
  seeking and loving what is true and good."7
  
  1705 By virtue of his soul and his spiritual powers of intellect and
  will, man is endowed with freedom, an "outstanding manifestation of
  the divine image."8
  
  1706 By his reason, man recognizes the voice of God which urges him
  "to do what is good and avoid what is evil."9 Everyone is obliged to
  follow this law, which makes itself heard in conscience and is
  fulfilled in the love of God and of neighbor. Living a moral life
  bears witness to the dignity of the person.
  
  1707 "Man, enticed by the Evil One, abused his freedom at the very
  beginning of history."10 He succumbed to temptation and did what was
  evil. He still desires the good, but his nature bears the wound of
  original sin. He is now inclined to evil and subject to error:    Man
  is divided in himself. As a result, the whole life of men, both
  individual and social, shows itself to be a struggle, and a dramatic
  one, between good and evil, between light and darkness.11


References:  (Gaudium et spes)
7 GS 15 #2; 8 GS 17; 9, GS 16; 10 GS 13 #1; 11 GS 13 #2.