...there was no agreement not to expand NATO. Sure there are no formal treaties, memo's etc. But....
https://nsarchive.gwu.edu/briefing-book/russia-programs/2017-12-12/nato-expansion-what-gorbachev-heard-western-leaders-early
Not once, but three times, Baker tried out the not one inch eastward formula with Gorbachev in the February 9, 1990, meeting. He agreed with Gorbachevs statement in response to the assurances that NATO expansion is unacceptable. Baker assured Gorbachev that neither the President nor I intend to extract any unilateral advantages from the processes that are taking place, and that the Americans understood 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 NATOs present military jurisdiction will spread in an eastern direction. (See Document 6)
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And at the time, it was unthinkable of a Warsaw Pact nation joining NATO. But Baker, not Bush made the commitment. But it was also implied NATO would become less of a Military alliance and the Soviet Union would become more integrated with Europe. But as the link show:
But inside the U.S. government, a different discussion continued, a debate about relations between NATO and Eastern Europe. Opinions differed, but the suggestion from the Defense Department as of October 25, 1990 was to leave the door ajar for East European membership in NATO. (See Document 27)
This would no doubt include Cheney, Rumsfeld, etc.