This is a common misconception. It's something that was discussed orally and taken out of its context and is being repeated today [1]. Besides the fact that NATO is a voluntary alliance and not an empire. At one point Russia was even considered for joining NATO.

The Budapest memorandum however was a written agreement, much more meaningful, though sadly not binding.

[1] https://hls.harvard.edu/today/there-was-no-promise-not-to-en...

> USSR was also once told in ~1990 “no NATO expansion to your doorstep”.

No, they weren’t, as even Gorbachev admitted, and even if they had been given private assurance not memorialized in a treaty or even a formal executive agreement, there would be no reasonable expectation of a binding commitment that survived the administrations on each side then in power, much less an actual collapse of the Soviet Union.

It’s not like the Soviet Union didn’t understand the mechanics of how binding international commitments are formed.