Hi there, I do not have an answer around 'the Project for the New American Century" as my political experience is more UK-centric.

However, I have recently written around this topic in regards to how US approaches to the Iraq War did not meet British PM Tony Blair's threshold for legitimate armed intervention. I believe I have some answer as to why Rumsfeld defaulted to the position you suggested.

The Bush administration, particularly in response to the September 11th attacks on the Twin Towers, advocated a position on national ‘self-defence’ which drastically altered the system of international relations. The 2002 National Security Strategy, was the culmination of what is more generally referred to as ‘The Bush Doctrine’; ‘a [US-centric] view towards the widening of the conventional right of self-defence so as to include taking traditionally unlawful uni-lateral pre-emptive military action against the ‘new’ threats of global terrorism and weapons of mass destruction.’

Although rhetorically expressed as directly intertwined with the aftermath of September 11, the notion of preventative wars was a topic of debate for ‘The Vulcans’, senior campaign policy advisors to the Bush electoral campaign, as early as the Spring of 2000.

Corroborated by both Professor Clifford Kiracofe, Virginia Military Institute, and US Treasury Secretary Paul O'Neill: Stephen Hadley, Deputy National Security Advisor 2001-2005 and National Security Advisor 2005-2009, briefed a group on this position in Spring of 2000.

Attendees included: his immediate superior at the National Security Council, Condoleezza Rice; Paul Wolfowitz, United States Deputy Secretary of Defence 2001-2005; Richard Perle; who would go on to become the Chairman of the Defense Policy Board Advisory Committee 2001-2003; Dick Cheney and former Secretary of State George Shultz. Hadley announced that the ‘number-one foreign policy agenda of the Bush administration would be Iraq and the unfinished business of removing Saddam Hussein from power.’

Similarly, in February 2001, the first meeting of the Bush National Security Council included briefings from Rice and CIA Director George Tenet on Iraq and sites that “might” be producing WMD’s. In response, Bush tasked the Secretary of Defence, Rumsfeld and Hugh Shelton, Joint Chief of Staff, to begin preparations for the use of US ground forces in the northern and southern no-fly zones of Iraq to bring down the Saddam regime.

Direct Sources:

(Christian Henderson, “The Bush Doctrine: From Theory to Practice”, Journal of Conflict & Security Law, 9:1 (2004) pp.5-6)

(W. Patrick Lang, “Drinking the Kool Aid”, Middle East Policy, 11:2 (2004) p.40)

Reading around the development of international policy approaches under the Bush administration:

Sean D. Murphy, “Terrorism and the Comcept of “Armed Attack” in Article 51 of the UN Charter” Harvard International Law Journal, 43 (2002)

Sean D. Murphy, “Contemporary Practice of the United States Relating to International Law” The American Journal of International Law, 96 (2002)

Raymond Hinnebusch, “The US Invasion of Iraq: Explanations and Implications,” Critique: Critical Middle Eastern Studies, 16:3, (2007)

Part of the context was that the Deputy Secretary of Defense Paul Wolfowitz had long considered Iraq under Saddam Hussein a major threat to the Middle East and to American interests, going back to the Carter administration in which Wolfowitz had served (yes, Wolfowitz was a Democrat prior to the election of Ronald Reagan, in whose administration he also served, along with that of George H.W. Bush). Wolfowitz was convinced that Saddam's Iraq, with its military power and its support of Palestinian terrorist group was a major threat to the Middle East's stability, especially after Iraq invaded Iran in 1980---unlike many American policymakers at the time, Wolfowitz thought that the US supporting Saddam's war against the new Islamic Republic of Iran (if mostly covertly and by proxy via the Gulf monarchies) would be a dangerous mistake, and that the end result would be Saddam strengthening his regional hand in the Gulf, thereby turning his aggressive ambitions elsewhere - which is exactly what happened in 1990.

Both the Carter Doctrine and the Reagan Corollary are important context here; by the 1980s the US was signaling that it was committed to defending the oil resources of the Persian Gulf from any outside aggression (which would be the USSR) or, the case of the new Iranian regime, any regional aggression against US allies (though this was unfortunately not clearly enunciated to Saddam until he invaded Kuwait in 1990). Wolfowitz was one of the voices singling out Iraq as an additional and, in his judgment, more threatening regional Middle East aggressor than Iran.

After the Gulf War of 1991 ended, Wolfowitz was one of those who wanted the first President Bush to not merely expel Saddam out of Kuwait, but destroy his regime altogether. Bush was very reluctant to do this however, since that was not part of the carefully crafted coalition plan, and he had neither authorization from Congress nor the UN Security Council for regime change in Iraq. Nevertheless, Bus he did publicly call on the oppressed Iraqi Shia and Kurdish groups to rise up against Saddam's regime - which they did, only to be slaughtered by his security forces, with no aid from the US. Wolfowitz was both horrified and deeply angered, and all throughout the 1990s and into the early 2000s disgusted with the fact that Saddam remained in power.

Wolfowitz shared these views with other officials in both the Reagan and first Bush administrations, including I. Lewis "Scooter" Libby, who worked with Wolfowitz in senior roles in the Pentagon during the first Bush administration under then-Secretary of Defense Dick Cheney, who he later hired Libby as both his chief of staff and national security advisor when Cheney assumed the Vice Presidency; Richard Perle; who served as an Assistant Secretary of Defense under Reagan while Wolfowitz was an Assistant Secretary of State, and who later chaired the Defense Policy Board, a senior advisory body to the Pentagon, during the second Bush administration; and Douglas Feith, who likewise served in mid-level to senior national security roles in the Reagan and Bush 41 administrations before he was made Undersecretary of Defense for Policy under Rumsfeld and Wolfowitz in the Bush 43 administration.

Among a relatively small number of other officials, most of them veterans of previous Republican administrations (and in a few cases, also Democratic ones like Wolfowitz), these are the people who are generally considered the "neo-cons" (neoconservatives) of the second Bush administration. All of them were hawkish on Saddam Hussein's Iraq, and many of them became key advisors in George W. Bush's administration, especially in the Department of Defense and the Office of the Vice President (Rumsfeld and Cheney, respectively).

So, going back to the original question - why did Rumsfeld ask about attacking Iraq? - the answer to that question is not easily deduced, but the most probable explanation is that he was influenced by both his advisers and those of his similarly hawkish protege Dick Cheney, along with Cheney himself (Rumsfeld had hired Cheney as a staff assistant way back in the Nixon administration, when Rumsfeld was serving in a domestic policy role, and Cheney became Rumsfeld's deputy when the latter was hired by Gerald Ford as White House Chief of Staff in late 1974).

There's a lot more to say about this subject, but that's a start. Here are some decent non-scholarly sources to get you started - I'll look for some scholarly ones that I can add to my answer, if you want.

Baker, Peter. Days of Fire: Bush and Cheney in the White House. Doubleday, 2013.

Draper, Robert. To Start a War: How the Bush Administration Took America into Iraq. Penguin Books, 2020.

Mann, James. Rise of the Vulcans: The History of Bush's War Cabinet. Viking Press, 2004.