This claim is disputable on a contextual basis, but legally false.
Arguments of NATO assuring Russia that no eastward expansion would occur are rooted in discussions surrounding German unification in 1990. In that meeting, former Secretary of State James Baker assured Soviet leader Mikhail Gorbachev that NATO held no plans to station troops in East Germany after unification, stating the following:
We understand that not only for the Soviet Union but for other European countries as well it is important to have guarantees that if the United States keeps its presence in Germany within the framework of NATO, not an inch of NATO’s present military jurisdiction will spread in an eastern direction. We believe that consultations and discussions within the framework of the “two + four” mechanism should guarantee that Germany’s unification will not lead to NATO’s military organization spreading to the east.
However, in the subsequent Treaty on the Final Settlement that cleared the way for unification, there was no mention of NATO.
Additionally, in a 1997 cooperation agreement between Russia and NATO, the following is noted among a list of shared principles:
respect for sovereignty, independence and territorial integrity of all
states and their inherent right to choose the means to ensure their
own security, the inviolability of borders and peoples' right of
self-determination as enshrined in the Helsinki Final Act and other
OSCE documents
As NATO has an "open door policy", states are within their right to request membership to the alliance. Assurances made by Baker were never officially agreed upon, and he did not have the authority to make those promises unilaterally.
There is also zero record of any "buffer zone" being discussed between NATO and Russia; military tensions had already cooled between the two by 1990.
