From the New Advent Catholic Encyclopedia ( http://www.newadvent.org/cathen/07163a.htm
(b) The question lies between the material, the metaphorical, and the symbolic sense of the word heart; whether the object of the devotion is the Heart of flesh, as such, or the love of Jesus Christ metaphorically signified by the word heart; or the Heart of flesh, but as symbol of the emotional and moral life of Jesus, and especially His love for us. We reply that worship is rightly paid to the Heart of flesh, inasmuch as the latter symbolizes and recalls the love of Jesus, and His emotional and moral life.
The whole explanation can be summed up in Catholic terms that the heart of flesh of Jesus is an icon of the love of Jesus and of His emotional and moral life.
But there is nothing to indicate that, during the first ten centuries, any worship was rendered the wounded Heart.
(2) It is in the eleventh and twelfth centuries that we find the first unmistakable indications of devotion to the Sacred Heart.
Also, the source states that the devotion as such only started, at earliest, in the 11th century, therefore after Rome left the Catholic Church.
It was in the fervent atmosphere of the Benedictine or Cistercian monasteries, in the world of Anselmian or Bernardine thought, that the devotion arose, although it is impossible to say positively what were its first texts or were its first votaries.
Finally, it affirms it was created exclusively within the jurisdiction of the Roman see.
Therefore we can say without any mistake:
1) Even if Rome were still in the Church this would not be a Catholic devotion, but a local one;
2) It is not something traditional in Christian asceticism and mysticism, but a medieval inovation;
3) Many Catholic mystics, Athonite elders and saints warn against over-emotional devotions and the kind of vision that originated this one;
4) The theological basis that the heart of Jesus is an icon of the love of Jesus has no basis on traditional iconography not in patristic writings where "heart" is the core of the being and not the center of its emotions, a symbology that is Western and medieval.
For all this, I think it is sure to say that the devotion, as it exists today, is not Catholic at all.
It could be if;
1) the heart acquired its traditional patristic sense of being the center of the being; emotions are together with rational thoughts in the concept of 'mind';
2) the human heart of Jesus was not mixed with this philosophical concept of heart;Not even when we have icons of abstract concepts (Hagia Sophia) we dissociate it from the whole being of Christ. Christ is Wisdom, God - the whole God - is love, not just His heart.
3) the heart of Jesus, the center and essence of His being would rightfully be represented by the icons of His life, Passion and Resurrection; at most expressed in His Uncreated Light, as a radiance of his Heart.
Finally, there is an issue that I haven't seen being brought up yet, but there is room for future heresies and schisms there. If Jesus has two natures, two wills, does He have one or two Hearts in the patristic sense of the word? Is there a center of His divine being and a center of his human being, or like these are united under one Person, it is this Person that has One Heart? And therefore, these two natures and wills are united over one Heart?