I think the definitive work is that of Hoyle & Fowler (1960).
They argued that supernovae were produced by two possible mechanisms - what they called an implosion/explosion or an explosion within degenerate matter. Both of these mechanisms required very high internal temperatures ($>2\times 10^{9}$ K) and they argued that this could only be achieved for both mechanisms if the star is more massive than the Chandrasekhar limit of $5.8M_{\odot}/\mu_e^2$, where $\mu_e$ is the number of mass units per electron (1 for hydrogen, 2 for helium, carbon, oxygen, 2.25 for 56Fe etc.); otherwise it could achieve a degenerate state, supported by electron degeneracy pressure, and cool indefinitely without contracting any further.
In addition, they successfully identified these two types of supernovae with Type I and Type II supernovae and argued that very high mass stars accounted for Type II supernovae via the implosion/explosion mechanism, whereas the Type I supernovae arose from the ignition of degenerate material in less massive stars, but still larger than the Chandrasekhar mass, which is $>1.2M_{\odot}$ for all plausible compositions. They even then conclude that stars up to $10M_{\odot}$ may lose enough mass to avoid a Type I supernova and become stable white dwarfs.
Though it seems not to be explicitly stated in the paper, it is implicit that the Sun could not explode as either type of supernova.
As an aside - I reminded myself of the plot of "Songs of Distant Earth" (I read it loooong ago). Clarke's story posits that the "neutrino problem" (the lack of detected solar electron-neutrinos), which was a thing back in the 60s, was actually because something odd was going on in the Sun's core. So I don't think that Clarke was suggesting that under normal stellar evolutonary circumstances the Sun would imminently explode. Indeed it was well known at that time that the Sun should continue on its main sequence life for billions of years.
