What should I major in for my BS if I am looking at doing BME in graduate school?

Initially I wanted to do Biomedical Engineering right off the bat in undergrad, because everything about it just fascinates me, I knew I wanted to do STEM, etc. But after a lot of research and receiving a lot of advice, I decided that doing something like BME (as opposed to ME or ChemE) for an undergrad degree wouldn’t be wise - in short, that it would be difficult to get a job unless I went on to grad school, which isn’t something I would have the money to do right after college.
Right now, then, I am trying to figure out the best major to go with for my undergrad degree. As silly as this sounds for someone who hopes to follow an engineering discipline at the graduate level, the biomedical aspect interests me more than the engineering (not to say I have no interest in the engineering, just that I am MORE drawn to the biological side of it). I also have a huge interest in neurology, so right now I’m registered at my college on the pre-med track (I will be a freshman in the fall). I know that those are two awfully different directions, but until I actually take the classes and experience the internships, I’m trying to leave both doors open. So my question, then, is not so much WHAT to major in, but whether I could do pre-med (say with biology or biochem) and still have the same chance at engineering in grad school, if it were biomedical engineering. If I could go to college for 8 years and do both I would!! :)) but my scholarship only lasts four years. Would a science major like those help me as much as an engineering major? What about if I minored in general engineering?

Thank you for any advice - I am trying to figure this out as early as I can, but my problem is just that I want to do everything!

Some universities have an undergraduate Biophysics major which works equally well for premed and for applying to BME grad programs.

If you want to keep the engineering option open, you’ll need more physics and math than a premed bio/biochem major would be required to take. Even if your school doesn’t have a biophysics major per se, looking at what these programs require might give you an idea of how to assemble a program that covers the same bases. https://phy.duke.edu/undergraduate/biophysics-major

That page was actually very helpful - thank you! I think if I include a minor in General Engineering, which my school offers, I think I should be able to satisfy enough physics and math classes to keep that option open. Would majoring in Biology, on the pre-med track, with a minor in General Engineering, however, be too much to take on? Having not been to college myself, of course, I don’t yet have a good idea of what would be getting in over my head.