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...