Can any stars ever form supermassive black holes?

As you say, making black holes quickly in the early Universe is a major unsolved problem in astrophysics. There are various hypotheses, of which two roughly correspond to supermassive stars. All basically involve trying to give the black hole a headstart in mass. There isn't really enough time to grow a $100,M_odot$ black hole to $10^9,M_odot$, so the idea is to rather get something more massive than a few $times1000,M_odot$. The most recent review I know of offhand is probably Volonteri (2010), but I'm not up to date on the literature.

Basically, imagine a few hundred thousand solar masses of gas collapsing into a primordial galaxy. As the gas collapses, it potentially fragments, depending on whether or not it can cool efficiently. If so, the fragments can presumably form stars, but in a very dense cluster. Ultimately, either (a) the massive stars in the cluster collapse into black holes that then merge into a larger black hole, which can subsequently accrete its way to supermassiveness (supermass?); or (b) the individual stars first merge in the centre, creating a star of perhaps some thousands of solar masses, which would collapse into a massive black hole that could grow.

If the primordial galactic cloud doesn't fragment, we expect a sort of monolithic collapse. Somewhere in the middle, gas will start to reach hydrostatic equilibrium: a protostar forms. But it's a protostar that's potentially accreting several solar masses of material per year (or even faster). So what happens next depends on whether that rapidly infall of material has to time to reach local thermal equilibrium with the protostar, or if it just piles up on the outside.

If the former, then the protostar can become very large: thousands or tens of thousands of solar masses, which, like the supermerger product above, presumably leaves a massive black hole in the end. If the small protostar evolves independently of the infalling gas, it's probably also big enough to leave a black hole, just a much smaller one: tens, maybe hundreds of solar masses. But it's a black hole embedded in this enormous cloud of infalling gas, which potentially settles into an envelope around the black hole. This structure has been dubbed a "quasi-star", and the black hole inside can grow very rapidly in this cocoon. Eventually, the envelope will evaporate/disperse, leaving the now massive black hole to continue to accrete its way to supermassiveness.

Note that these formation mechanisms are expected to be particular to the early Universe. Once you add even a small amount of metals to the gas, then star formation is expected to look much more like the "modern" Universe. In fact, star formation in these scenarios is still far from settled. The main reason is that you need to follow how the gas evolves from the scale of the protogalaxy all the way down to the protostars inside. This is a range of scales something like $1,mathrm{AU}/10,000,mathrm{ly}approx10^{-9}$, which is numerically very difficult.

And, finally, to answer the question directly, in neither of the supermassive star options do the stars really collapse in supermassive black holes. SMBHs are so big that they can only have grown so large through accretion. The supermassive stars would collapse into their progenitors (or "seeds"), which would subsequently grow to such large masses.