Not true, there was never any commitment from the US government not to support NATO expansion, and the right of independent states to enter into alliances of their choice was accepted by Gorbachev.
It started with Gorbachev and Bush, and back then the assurance could reasonably be said to apply to Germany, though the talks were broader -
from -
https://www.theguardian.com/world/2...ief-in-nato-betrayal-and-why-it-matters-today
'Not One Inch: America, Russia, and the Making of the Cold War Stalemate, by the prize-winning historian Mary Elise Sarotte, charts all the private discussions within the western alliance and with Russia over enlargement and reveals Russia as powerless to slow the ratchet effect of the opening of Nato’s door. The author concludes
the charge of betrayal is technically untrue, but has a psychological truth.
What is the basis of the complaint?
At one level it narrowly focuses both on verbal commitments made by the US secretary of state James Baker under President George HW Bush and the terms of a treaty signed on 12 September 1990 setting out how Nato troops could operate in the territory of the former East Germany.
Putin claims that Baker, in a discussion on 9 February 1990 with the Soviet leader, Mikhail Gorbachev, made the promise that Nato would not expand to the east if Russia accepted Germany’s unification.
The following day Chancellor Helmut Kohl, ambiguous about Germany remaining in Nato after unification, also told Gorbachev “naturally Nato could not expand its territory to the current territory of the GDR”. The promise was repeated in a speech by the Nato secretary general on 17 May, a promise cited by Putin in his Munich speech. In his memoirs, Gorbachev described these assurances as the moment that cleared the way for compromise on Germany.
...
Russia was given verbal assurances about the limits of Nato’s expansion, but no written guarantees. In March 1991 John Major, for instance, was asked by the Soviet defence minister, Marshal Dmitry Yazov, about eastern Europe’s interest in joining Nato. Major, according to the diaries of the British ambassador to Moscow, Rodric Braithwaite, assured him “nothing of that sort will ever happen”.'
Yeltsin's understanding is also somewhat moot, but this patriotic US website expresses it fairly I think -
https://warontherocks.com/2019/11/p...told-about-nato-in-1993-and-why-it-matters-2/
'What mattered most to Yeltsin was not what Gorbachev was told in 1990, but rather what he was told in October 1993: that the United States was pursuing a Partnership for Peace for all European countries rather than NATO membership for only some European countries. This was not a promise either, but it cemented for the Russians the narrative that regardless of what the United States claimed in conversations with their leaders, it would maximize the American position without regard for Russian interests....
Encapsulating all of the ambiguities of that period more than any other meeting was a conversation that took place in Moscow in October 1993. U.S. Secretary of State Warren Christopher had traveled to Moscow to explain in advance of the January 1994 NATO summit that the United States would not support new members joining the alliance, but would rather develop a Partnership for Peace that would include all states of the former Warsaw Pact. Yeltsin’s relief was palpable. He thought he had dodged the NATO enlargement bullet at a time at which he was in a raging political battle against hardliners at home. A year later, when he discovered that enlargement was not only on the table but would in fact be proceeding, Yeltsin was apoplectic, and he railed against Clinton publicly at a meeting in Budapest...
Thanks to the Freedom of Information Act, we have the declassified memorandum of conversation (MemCon), which sheds much more light on what was said in October 1993
The idea of partnership for all rather than membership for some was, said Yeltsin, “a great idea, really great. Tell Bill I am thrilled by this brilliant stroke.” [Warren] Christopher noted, “We will tell him that you bought his recommendation with real enthusiasm.”
According to the MemCon, it was only then that Christopher said that the United States would be “looking at the question of membership as a longer term eventuality.” We do not know whether Yeltsin or other Russian officials in the room reacted to this point, nor do we know how clearly Christopher delivered this message.
In the MemCon, this specific point is not in quotation marks as is the case with a number of the other comments that were reported.'
(my emphases)