Putin brought it up again early in his Presidency as a hypothetical. I think his behavior within in the alliance would have been *distinctive*, a super-Gaullism, x10, forcing the US to use smaller 'coalitions of the willing' to do much of anything proactive or potentially controversial.
From the American side, James Baker proposed an invitation to Russia around 2001-2002. He was extremely influential in the diplomacy of the first Bush administration. Don't know if he had similar views in the 90s and would have pushed them in a 2nd George HW Bush term, and as Elektronaut pointed out, Yeltsin reciprocating may not be possible.
In any case, although James Baker was a family friend and vital to George W. Bush securing the Presidency in 2001, that was pretty much the end of his influence under Bush the Younger, he had no role shaping the latter Bush's foreign policy, yielding place to Dick Cheney and Don Rumsfeld and Paul Wolfowitz (Condoleeza Rice and Colin Powell were appointed to high positions and tried to influence policy, and perhaps exerted some early influence, but in the event of any differences of opinion - Cheney, Rumsfeld, Wolfowitz or Bolton always won - at least until Bob Gates was appointed in late 2006/7).