I suppose it all depends on what you mean when you're talking about a clan, and how far you're willing to push the comparisons between social structures in different times and places. The short answer would be that yes, you can find parallels to the clan system in England (if you squint), but they're best found in the 1400s and early 1500s, not in the early to middle Anglo-Saxon period (i.e. post-Roman, pre-Alfred). I should add the caveat here that I'm primarily an early and central medievalist, so all of the info on the late medieval/early modern stuff here is hazy at best, and I'd appreciate anyone with more knowledge correcting me!

The first thing to note is that, just like technological advances, social and economic development do not follow a tech tree. It's not so much 'tribe' to 'clan' to 'kingdom' to 'state' as a mish-mash of aspects of all of these, concurrently, with increasing and decreasing complexity based on many factors. So finding parallels to the Scottish Clans of the 15th to 18th centuries are not a question of "going back far enough" to an earlier point on the tech tree but more likely to be found in contemporary neighbours.

As I understand it, the clan system was a kin-based system of social organisation, where free men and their families were bound to clan leaders through bonds of obligation and solidarity. Not all members of a clan were related by blood, but clans were small-scale units and there would have been frequent marriage between the clan elite. This elite was supported through tribute and booty from raids and wealth was redistributed via gifts and salaries paid by the chief to his retainers and warriors. The system originated in the late 14th century across the northern Gaelic cultural area (modern Ulster and Connacht in Ireland and the Highland and Islands of Scotland) and survived until 1603 in Ireland and the mid 1700s in Scotland. The origins and workings of this system in Ulster are detailed in Katherine Simms, 'From Kings to Warlords'. I wish I could provide something similar for Scotland, but I'm not really familiar with the field, although IIRC Simms talks about the connections with Scotland and probably has further reading.

So was there anything comparable in pre-Viking Anglo-Saxon England? Not really. England in the period 500-900AD was similarly dominated by small polities, but these were kingdoms, not kin-based chiefdoms and by about 600-650AD would have been considerably more structurally complex than a Scottish clan, particularly in eastern and central England. They had kings and an aristocracy, which derived its income not only from tribute but also from landowning. Kin was important in Anglo-Saxon England, but did not play nearly as much of a central role as it did in Early Modern Scotland. The best brief introduction to this period is in Chris Wickham's 'The Inheritance for Rome', chapter 7, which has lots of further reading.

As for a possible English parallel to the Clan system, my first thought would be the so called 'Bastard Feudalism' system of the Later Middle Ages in England. This was, in essence, a system of paid retainer-warriors centered around aristocratic households. These were much bigger than clans however, were based fundamentally on land rent rather than tribute and had little significant kin-ties in them. This system disappeared in the 1500s under the more centralised Tudor court. My information here is really just remembered from undergraduate lectures though, so if any specialists on this period are around and could elaborate I'd very much appreciate it!