Here is a sampling of views on time that you'll find in recent literature:
Presentism: Only the present moment is real; neither the past nor the future exist. Among other things, this view is supposed to help make sense of our sense that time really passes, but it does that (to the extent that it does) by treating the present as an infinitely thin slice and not as something eternal.On the contrary: the present is utterly evanescent.
The Block Universe: On this view, all events exist eternally, as it were. That includes all events before my writing this post, all events simultaneous (in some frame of reference) with my writing, and all events after my writing. This view is claimed to fit best with the understanding of time that Relativity provides, but there is no eternal now. "Now" has no metaphysical interest whatsoever. "Now" is like "here": it's what is sometimes called an indexical term, picking out what it picks out only relative to its use. When you say "I am here" and I say "I am here," what you are saying is not what I am saying. You use "I" to refer to yourself; I use it to refer to myself. You use "here" to refer to where you are; I use "here" to refer to where I am. But there is no privileged "I" and no privileged "here." There are just speakers and locations. On the block universe view, "now" works the same way. It doesn't pick out anything special or privileged or metaphysically distinguished. On the block view, there are temporal relations (earlier than, simultaneous with, later than) but no special times and no unique, let alone eternal "now."
The Growing Block: This view takes seriously the worry that that really are facts about the past, and that if Presentism is true, this is hard to understand; on Presentism, the past does not exist. But this view also takes seriously the idea that time really does pass, and that it's hard to reconcile this with the Block Universe. So the Growing Block view says that the totality of existence keeps growing. The past is real, and that's why there can be truths about it. The present is continuously slipping into the past, but is indeed the present for an instant. The future doesn't exist, but will come into being, and once in being will not cease to be. Once again, however, no eternal now.
There is another view that's received some attention recently. It's called fragmentalism. Insofar as Presentism allows for truths about the past, there is an argument due to John McTaggart Ellis McTaggart (who wins the prize for nifty names) that time comes out contradictory on that view. In the past it was true that I am now in Canada. In the present it is true that I am not now in Canada. But the Presentist also wants to say that the word "now" is not like the word "here." It's not an "indexical" term but a way of picking out something metaphysically privileged. And so we apparently get a contradiction, which McTaggart takes to show that the view must be false. Fragmentalism embraces the contradictions but segregates things into "fragments" that are internally consistent but mutually inconsistent. We are never in a position to reason from more than one fragment, and so the worst logical consequences of inconsistency are avoided. But there is still no one eternal "now," though there may be some sense (I'm genuinely not sure) in which the fragments have something like eternal status.
There is also a view that some physicists take quite seriously, according to which time is not a fundamental feature of the world at all, but is "emergent" from more basic physical structure. This may (or may not) mean positing a fundamental and timeless reality, and in that sense might be sorta, kinda like an eternal "now." But since "now" is a temporal notion and time emerges out of the fundamental reality on this account, "eternal now" sounds like a bad way to put the idea.
So... there are views according to which there's no past or future, but the now—the present—isn't eternal. And there are views on which there's no "now," except in a metaphysically uninteresting sense, but there is something like an eternally (timelessly?) real collection of events. And there are a few more exotic views as well, but none that clearly fit the model of an eternal "now." I'm not sure whether any of these views are what you have in mind.
There is also, however, a more-or-less mystical notion that one sometimes runs into. One might think of it as dealing with how things are from God's timeless point of view and our capacity to get occasional glimpses of that. (I'm neither endorsing nor rejecting the idea.) The language of an "eternal now" is often used to capture the idea. The theologian Paul Tillich published a collection of sermons titled The Eternal Now. That said, though I have a soft spot for Tillich, I don't really see much connection between this idea of an eternal Now and the concerns that come up in contemporary philosophy of time.