Well I shot an elephant so now I'm an expert! (not) With a 375, use a scope, I explain why down below.
If buffalo and especially leopard is in your future, or any other dawn or dusk or dark shadows critters (I used mine on a jackal at dusk)... Consider an illuminated reticle. For the money, I love my Leupold VX6 2-12 on the 375, admittedly, don't really need that 12X.. But it can be handy if trying to get a good look at something. A VX6 in 1-6 would be awesome and practical (90% sure I will put one on my 416)... but both are way out of your budget (sorry).... However don't buy two scopes. with all the options available these days, no need t oscrew around and you will be way better off to put the extra money into a little better quality. I also really like the 30mm tubes on a big bore gun where weight is not much concern. Just go look through them. Before you buy, go look at a Leupold VXR in 2-7 and one in 1-4. If on a budget, I would do a 2-7 (even for elephant) on a 375 and the 1-4 on a 416 or bigger.
If you want to shoot an elephant with a 375, use a scope!!! If you really want to shoot open sighted, use a gun with the power of a 458 Lott or more, just to get that extra shock and knock down if you are off target a bit. People kill elephants with 375's all the time, but there is so much that is uncontrollable and that you just don't know until after the bullet hits, have some "extra" on accuracy or on "thump", you might be ok giving up a little on one or the other, but not both... I suspect most PH's will tell you they prefer the accuracy and to bring a gun you shoot well with a good scope.
I'll pass on what I thought was sensible advice. A 375 has excellent penetration with the right bullet and load. (I was terribly impressed by the Federal Premium Hydro Solids, blew right threw the elephant's head, it was a 416 Rem Mag but I'm sure a 375 would do the same or more) 375's are inherently accurate and again with proper loads, have great trajectory on par with a 30-06. It is truly the best all around African caliber! So good choice.... It will no doubt drop an elephant of any size, in it's tracks, "IF" you hit the brain. USE A SCOPE!
If you want to shoot an elephant open sighted, use something in the range of a 458, preferably a Lott or with the best bullet and load possible if a Win Mag. Or as a bare minimum, a 416. The three 416's I'm familiar with are the Rem Mag which I used, the Ruger which duplicates it, and the Rigby which I also own now but have not used on an animal.. but I do think the Rem Mag was designed to duplicate factory Rigby loads. Anyway, back to the 375 H & H (or Ruger), these 416's have 20% more energy than the 375's... So you have that much more "whack" if you miss the brain by a 1/2 inch.... I missed the brain by about that but also grazed the spine where it connects and nocked him down and then put in two more (always pay the insurance unless you PH stops you). One through the chest (top down or bottom up) and one in the top of the head. I'll never know if that bull would have dropped if hit in the same place with my 375. But I'm positive it would have with a 458.
Ok enough preaching; Stay at 2x or lower on the low side and a minimum of 4x on the high side, but personally I would go to 6 or 7x on the high side. 1-4, 1.5-5, 1-6, 2-7, 2-10, or 2-12... All good on a 375. I have no experience with a Nikon on a big bore but the cheap one I had was accurate and decent, just not top end. I have a Swarovski Z6i in 2-12 and in my opinion, it is not worth the extra cost over a VX6. Love the z5 line and the Zeiss HD5 line but again, on a DG rifle, I like the 30mm tube. I have Warne QD Med rings on my 416 rem mag and the tightest point is the bolt handle to the eyepiece. Yes I can see the sights at 1 or 1.5 X and a shadow at 2x but don't ever recall seeing that when I was shooting at an animal. I completely agree with Royal, mount as low as you can... This might be a reason to go to a very small scope like a VX3 in 1.5 to 5 or something like that.
I know you said your budget is only so much, but the old adage of spending 1-2 times as much on a scope as on the rifle is not far from being great advice.... To me the VX3 or VXR would be as low a quality as I would go. Monarch 3 or better in a Nikon? I'm not real familiar?