There is much information available, but in a short summary I think if you read the below text which is a post on the subject matter from Matthew Steenberg of monachos.net who is at Oxford furthering his studies you'll get the general idea. Steenberg is very objective and non-prejudiced in many of his commentaries at least from my perspective.
"Some of the most straightforward remarks in this regard actually come down to us from St Gregory the Great, pope of Rome in the sixth-seventh centuries A.D. His remarks were occasioned by the Emperor's application of the title 'Ecumenical Patriarch' to St John the Faster, Patriarch of Constantinople. Gregory was deeply disturbed by this title, not because he felt that its attribution to a patriarch other than that of the see of St Peter was improper, but because he felt the very notion of such a title or rank was incorrect (we must keep in mind that St Gregory understood 'ecumenical' to mean universal in authority and power, which is not how the term as it is used in the title 'ecumenical patriarch' has come down to us today'). In response to this, St Gregory writes to St John:
"Certainly Peter, the first of the Apostles, himself a member of the holy and universal Church, Paul, Andrew, John -- what were these but heads of particular communities? And yet all were members under one Head [...] the prelates of this Apostolic See [i.e. Rome], which by the providence of God I serve, had the honor offered them of being called 'universal' (oikoumenikos) by the venerable Council of Chalcedon. Yet not one of them has ever wished to be called by such a title, or has seized upon this ill-advised name, lest if, in virtue of the rank of the pontificate he took to himself the glory of singularity, he might seem to have denied it to all his brethren [...]"
(Excerpted from Book 5 of the collected epistles of St Gregory the Great of Rome, Epistle 18).
Later he writes in a similar vein:
"This name of Universality was offered by the Holy Synod of Chalcedon to the pontiff of the apostolic see which by the Providence of God I serve [i.e. the see of Rome]. But no one of my predecessors has ever consented to use this so profane a title since, forsooth, if one Patriarch is called Universal, the name of Patriarch in the case of the rest is derogated. But far be this from the mind of a Christian that any on should wish to seize for himself that whereby he might seem in the least degree to lessen the honor of his brethren..."
(Book 5, Epistle 43)
When, a short time later he writes to the Emperor (Maurice) on the matter, he is yet more emphatic:
"Now I confidently say that whosoever calls himself, or desires to be called, Universal Priest, is in his elation the precursor of Antichrist, because he proudly puts himself above all others."
(Book 7, Epistle 33)
Later, he writes to the Bishop of Alexandria (Evlogios):
"Your Blessedness [...] You address me saying, 'As you have commanded'. This word 'command' I beg you to remove from my hearing, since I know who I am and who you are. For in position you are my brother, in character my father. [...] In the preface of the epistle which you have addressed to myself, who forbade it, you have thought fit to make use of a proud appellation, calling me Universal Pope. But I beg you, most sweet Holiness: do this no more, since what is given to another beyond what reason demands, is subtracted from yourself [...] For if your Holiness calls me Universal Pope, you deny that you are yourself what you call me universally."
(Book 8, Epistle 30)
There are multiple things of note in these quotations, but among them I might simply point out Gregory's own insistence that, prior to his own day (he reposed in A.D. 604), no bishop of Rome had ever claimed episcopal primacy of authority."