To my knowledge (if it has any value), there is no such modern theory without pairing.
So the question is why, right ?
There is no proposal trying to explain high-temperature superconductors (HTS) without paired electrons for the simple reason that there are (at least supposed, let me know if you disagree) evidences for pairing ! If you know about the Aharonov-Bohm effect, you will not be shocked by two facts
due to Aharonov-Bohm, there are some quantities which oscillates at the ratio $\Phi /2 \Phi_{0}$, with $\Phi_{0}=h/2e$ the superconducting flux quantum, $\Phi$ being the magnetic flux piercing some non-connected region of matter (metal or superconductors in our case of concern). So, for normal electrons some things actually oscillate with the characteristic value of the Planck constant $h$ over the charge of the electron $e$
there is no known quantity oscillating at $\Phi /2 \Phi_{0}$ for superconductors, everything is oscillating at $\Phi / \Phi_{0}$, even for HTS, as for the Little-Parks effect for instance. In short, Little-Parks effect is the oscillation with the magnetic flux of the critical temperature of a superconducting plate with a hole in the middle. All these interfering experiments need the composite particle with charge $2e$ to be understood.
Unfortunately, I can not find some good references discussing this point. I'll try to include some latter. I believe there are some discussions in the book by A.J. Leggett Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems Oxford Graduate Texts.
edit Haha edit ! I've done the previous answer based on some discussions I had with colleagues. I was (stupidly) looking for Little-Parks effect + HTS on Google Scholar without result :-( Of course the good key words to find something are flux quantisation + HTS. The first result is already enough I believe
Gough, C. E., Colclough, M. S., Forgan, E. M., Jordan, R. G., Keene, M., Muirhead, C. M., Rae, A. I. M., et al. Flux quantization in a high-Tc superconductor. Nature, 326(6116), 855–855 (1987).
Nevertheless, the Leggett's book (that I read when it was first published and forget then) should contains a lot of material in one of its last section. Leggett discusses there all what we know about cuprates.