This is a difficult question to answer, as there is certainly no evidence of official contact between England and Byzantium. However, there are oblique hints that people and envoys were on the move between the two opposite ends of Europe. Since the time of Aethelstan (924-39), for example, English kings sometimes described himself as the basileus 'of the English' or basileus 'of all of Albion' - βασιλεύς being the Greek word for 'emperor'. Sarah Foot's recent biography of Aethelstan (2011) further suggests that the king's awareness of Greek norms can be tied to Duke Hugh of the Franks' embassy to the king, as he sent a number of gifts to England in 926, which included a Byzantine onyx vase. Similarly, a fragment of the True Cross given to Aethelstan in the same collection may have had an eastern origin as well, since the Byzantines gave a similar relic to the Western Franks in 872. This is only a supposition based on what we know about - I think it would be safe to presume that the Anglo-Saxon elite were fully aware of the implications of the word basileus even without material evidence of Constantinople's wealth in their treasury. Their kingdom was after all increasingly dominant in Britain, so a title as grand as 'emperor' was eminently suitable - though we should note that other formulations were also used and that basileus was not used as the title of the king in all cases.

Aethelred (978-1016), as the heir of the House of Wessex, continued the same trend and based on a brief look at the Prosopography of Anglo-Saxon England, I see that he was described as basileus in 50 of his charters (or there about), compared to 7 featuring imperator, the Latin word for 'emperor'. Admittedly the usage of this word was dwarfed by the much more common 'king of the English', but it is nonetheless I think a significant demonstration of the pretensions of Aethelred's court. The same pattern would continue into the reigns of Cnut and Edward the Confessor, which tells you a little about how imperial nomenclature (and eastern influence) was hardly unknown in late Anglo-Saxon England.

There are a number of other interesting connections. William of Malmesbury, a twelve-century historian, recorded that there was a Greek monk at Malmesbury named Constantine in the early eleventh century. More surprisingly, William also said that the monks believed that Constantine was once an archbishop, but did not know why he left his native land. This tale may well be legendary, but it does tell us something about what later generations thought about the possibility of foreigners in eleventh-century Malmesbury. Sometime between 964 and 1030, we also know that a Greek monk named Andrew served in the New Minster at Winchester, since he was named in the local Book of Life, which recorded names that were to be commended to God during mass. Finally, we also have a certain 'Bishop Sigewold', apparently a 'Greek by origin', who was involved in a property dispute in the 960s. There is a very remote possibility that he can be identified as Nikephoros (both this and Sigewold means the same: 'Victory-Bearing') of Herakleia, who was apparently snubbed by the Byzantine emperor in 956 for unknown reasons, as recorded by the historian John Skylitzes. This is a very unlikely suggestion, but at the very least there was someone in England of Greek origins but remembered with an Old English name, who was presumably wealthy enough to argue with another member of the elite (a Dane, coincidentally enough) about land. Though ultimately still obscure, especially as we can't place Sigewold in a known bishopric at this time, this existence is still telling and is an important reminder that bishops can be a diverse bunch. It is worth remembering that during the 'golden age' of the Anglo-Saxon church in the late seventh century, as seen by the Venerable Bede, it was in fact led by Theodore of Tarsus, a Greek from Cilicia (whose right-hand man was Hadrian, an abbot from North Africa), and there were more than a few Irish and Frankish bishops back then too.

Anyway, these are the people who we do know from around Aethelred's reign with possible eastern connections. But I think we should still be cautious when analysing these pieces of evidence. Aside from the doubt I expressed above, we should note that being Greek by origin did not mean that people really saw themselves as 'Greek' or that they had travelled from Byzantium. Identity is a difficult issue to understand and often very flexible - someone with an ethnic name, for instance, did not necessarily have to belong to that ethnicity (or their 'original' ethnicity, for that matter). This is why we have to supplement this with other evidence. Luckily for us, scholars such as Michael Lapidge have done all the legwork already. For one version of the Anglo-Saxon Chronicle, the Greek word dromon was used to refer to ships rather than an Old English word, similarly when the chronicler referred to light galleys as moneris, a Latinised form of μονήρης. As the author was Aethelweard, a counsellor serving King Edgar (959-75), it is possible, as Jonathan Shepard has recently suggested, that it was a result of the court's acquaintance with Scandinavians who had served in the Byzantine navy. Other Scandinavian links with England, which are well-known, may well have led to the donation of relics of the eastern saints Cosmas, Damian, and George to Winchester by Emma, wife of Cnut (and previously of Aethelred) after the king's death. I find this a bit unlikely, since all three saints were well known in the west before this period, but it's still something to be considered.

What about the Byzantines then? What evidence exist of their views of the Anglo-Saxons? Michael Psellos, the premier intellectual of the eleventh century, noted c.1060s that life amongst provincials was akin to be a Greek exile living in Britain, implying that the provincial Greeks did not speak the language properly (or at least, not up to the high standards of Psellos). This perhaps also hints at his awareness that people in Britain spoke poor Greek. A Byzantine seal of a treasury official from the same time has also been found in London and the evidence has even been interpreted by Peter Frankopan as signs that the Byzantines were recruiting Anglo-Saxon mercenaries, well before the First Crusade. This is not the only example, as the seal of John Raphael, the commander of the Varangians in Italy in 1046, has also been discovered in Winchester, likewise for a seal of the logothete Leo in London, c.1000. We do know of Anglo-Saxons going east to serve in the Varangian Guard after the Norman Conquest of 1066 (the earliest evidence is perhaps the words of Kekaumenos, who got annoyed that foreigners 'coming from England' were honoured c.1070s, though note that this reading of the manuscript is disputed), so it's not a stretch to think that adventurers from Britain were serving in Constantinople before then as well.

My last suggestion is just that, a suggestion, but I hope that my summary of the available evidence does not rule out that possibility. We definitely do not have any official accounts of diplomatic contact, but from other sources, whether the odd names that popped up in the sources or the occasional references, I think it would be fair to say that the Anglo-Saxon court of Aethelred knew about Byzantium and plausibly had some contact with it, if only very tangentially through intermediaries.