UPDATED, 2:30 p.m.: Senate Majority Leader Harry Reid is making a formal pitch to Speaker John A. Boehner, R-Ohio, for a broader budget conference.

The Nevada Democrat continues to insist, however, that Boehner put the Senate’s “clean” continuing resolution on the House floor to get the government back open as a precondition.

“I propose that you allow this joint resolution to pass, reopening the Government. And I commit to name conferees to a budget conference, as soon as the Government reopens,” Reid wrote in a Wednesday letter. “That conference can discuss the important fiscal issues facing our nation.”

It comes in advance of a scheduled meeting of the top four congressional leaders with President Barack Obama at the White House.

Reid said in the letter that he viewed the conference as a good venue for broader debates on discretionary spending.

In the letter, Reid compared the situation facing Boehner and the House GOP to his own decisions over continuing funding for the Iraq War after Democrats reclaimed control of the chamber in 2006. At the time, Democratic leaders were under intense pressure from liberals to defund the war to force its end.

“In those days, when President Bush was Commander in Chief, I could have taken the steps that you are taking now to block Government funding in order to gain leverage to end the war,” Reid wrote. “I faced a lot of pressure from my own base to take that action. But I did not do that. I felt that it would have been devastating to America.”

Boehner quickly rejected the offer, however. His spokesman Michael Steel released a statement, saying, “Offering to negotiate only after Democrats get everything they want is not much of an offer. Today, the House will continue to pass bills that reflect the American people’s priorities. The Senate passed the troop funding bill this weekend — will they now say ‘no’ to funding for veterans, our National Parks and the National Institutes of Health?”

In either case, there’s no guarantee Reid could follow through on naming conferees, however. Motions to go to conference are debatable on the Senate floor (which means they are subject to filibuster).

Senate Democrats tried 18 times to get to a conference on a House-Senate budget resolution, but Budget Chairwoman Patty Murray, D-Wash., and others faced persistent GOP objections over concerns that such a conference could create a glide path for raising the debt limit.

UPDATE: Boehner spokesman Michael Steel released the following statement in response to the Reid letter: “The entire government is shut down right now because Washington Democrats refuse to even talk about fairness for all Americans under ObamaCare. Offering to negotiate only after Democrats get everything they want is not much of an offer. Today, the House will continue to pass bills that reflect the American people’s priorities. The Senate passed the troop funding bill this weekend — will they now say ‘no’ to funding for veterans, our National Parks and the National Institutes of Health?”

Meanwhile, WGDB asked three of the four Senate Democratic leaders why their preference is to move to a budget conference at this point, as apposed to an appropriations one. Reid, Sen. Richard J. Durbin, D-Ill., and Sen. Charles E. Schumer, D-N.Y., all seemed to think that the focus should be on the oft-rejected budget conference so that negotiators could set a top-line spending number. But given that the government already is shut down, that the pending clean CR is only for six weeks and the Democratic hope is that a longer-term omnibus can clear Congress before Christmas, it seems like fighting futilely for weeks over spending levels the two sides might never agree on would not be the most effective use of time, even if Senate Democrats could move to appoint conferees without being blocked.

“That’s where the budget conference takes you, you agree to a top line figure… then the Appropriations subcommittees can do their work,” Durbin said. “[A budget conference] is broader. First, an appropriations committee can’t really go to conference because we don’t really have a bottom line. We mark to a different number than the House. So the budget committee would work out that number. That’s the starting point.”

Meredith Shiner contributed in updating this report.