Yes, a few times, in the 13th century and then again in the 15th.
The two biggest obstacles to overcome were papal authority and the “filioque” clause. These were the two issues that caused the schism in 1054.
For papal authority, the pope claimed that since the church was founded in Rome when the emperors still lived there, the pope (as bishop and patriarch of Rome) had authority over the entire Christian church. The patriarch of Constantinople, of course, disagreed; spiritual authority had moved to Constantinople along with the emperor.
The “filioque” clause refers to the statement “and from the son”, which had been added to the Nicene Creed by the Roman church in the 11th century. This has to do with where each part of the Trinity comes from. Did the Holy Spirit come from God alone, as the Council of Nicaea had concluded (way back in the 4th century), or did it come both from God “and from the son”, as Rome argued? There were other issues as well (among other things, the Latins used unleavened bread for the Eucharist, while the Greeks used leavened bread) but those were the two major ones.
In 1204, the Fourth Crusade captured Constantinople. The crusade was originally meant to go to Egypt, and the fact that it ended up attacking and conquering a fellow Christian empire was, even at the time, a big controversy to say the least. The pope, Innocent III, was certainly interested in reuniting the churches, but that wasn’t exactly what he had in mind…still, now that the empire had been conquered, he thought it was a good opportunity. The Greek church wasn’t completely suppressed, but if Greek priests and bishops wanted to keep their positions, they had to swear allegiance to the pope. Most refused, so there was never any real “union” at this point. In 1261, the crusaders were defeated by Emperor Michael VIII, who had been ruling in exile not far away in Nicaea.
Michael knew that the Latin west would simply try to take Constantinople back again, and one way to stop them would be to negotiate a union of the churches with the pope. At the Council of Lyon in 1274, Michael’s representatives agreed to accept Roman authority and the Filioque clause. But virtually no one in Constantinople was interested in this at the time. Even Michael knew it was just a temporary distraction. The union was ignored while he was alive, and officially repudiated after he died in 1282.
The next attempt came about 150 years later. By 1438, the Empire was pretty much all gone. The only territory left was Constantinople and its immediate surroundings. The Ottomans had unsuccessfully besieged Constantinople in 1422, but without help from the west, it was obvious that it was only a matter of time before Constantinople fell as well. The Greeks also knew that help would only come on the condition that the churches were reunited.
In 1438, 700 delegates from the Byzantine Empire, including the emperor John VIII, arrived at the Council of Ferrara. This is sometimes called the Council of Florence (or Ferrara-Florence), because it was moved to Florence in 1439 when plague broke out in Ferrara.
The Greek delegates debated the Filioque and Roman authority with their Roman counterparts. But they also disagreed with each other. Why should they be the ones to unite with Rome, when it was the Roman church that was in schism with Constantinople, and had maybe even fallen into heresy? But the pro-Roman faction simply observed the reality that the Greek church was in no position to make demands. They needed help and uniting the churches was the only way to do it. The pro-Romans felt their differences could easily be overcome.
So in May 1439 they agreed to the union. They accepted the Filioque clause, but did not have to add it to the Greek version of the Nicene Creed. For other minor matters, such as leavened vs. unleavened bread, both sides agreed that they did not have to change their respective customs. But they couldn’t compromise on the authority of the Pope. The patriarch of Constantinople simply had to recognize that he was now second place in the church hierarchy. The union was officially celebrated in Florence on July 6, 1439, and the Greek delegates then returned home to Constantinople (although some of them stayed behind to study and teach in Italy).
A few years later in 1443-1444, the promised assistance from the west arrived - an army led by Hungary and Poland, as well as a a fleet of French and Hungarian ships that tried to sail past the Ottomans into the Black Sea. But the fleet was stopped, and the army was destroyed at the Battle of Varna in Bulgaria in November 1444. That was the last time a major crusade was launched. Help continued to trickle into Constantinople over the years, but not enough to make any difference.
In Constantinople, the union was also not accepted by most people. Union could be achieved in abstract theological terms, but there was no way to enforce it. Churches remained empty, even the cathedral of Hagia Sophia. Supposedly some anti-union citizens remarked that “it would be better to see the sultana’s turban in Constantinople than the Latin mitre.”
Emperor John died in 1448 and was succeeded by his brother Constantine XI, who also supported the union. But Constantinople finally fell to the Ottomans in 1453 and the church union died along with the empire.
Sources:
Jonathan Harris, The End of Byzantium (Yale University Press, 2012)
Joseph Gill, The Council of Florence (Cambridge University Press, 1959)
Robert Browning, The Byzantine Empire (Catholic University of America Press, 1980, rev. ed., 1992)