Speaker John A. Boehner’s proposed fallback plan to deal with the tax rate increases looming at the end of the year would give Senate Majority Leader Harry Reid what might be a much-needed Christmas present.

When the House Rules Committee gave formal notice of a meeting to set up floor debate for test votes on two proposals to extend some expiring tax rates, the panel’s advisory noted the measures would be offered as substitute amendments to a carefully selected but wholly unrelated bill.

The shell of a joint resolution that began as a routine annual extension of import restrictions on Myanmar will become the legislative vehicle selected for the “plan B” by Boehner, R-Ohio. The measure (H J Res 66) has already passed through both the House and Senate in different forms, which may give it obscure procedural advantages that could cut down on the time needed to approve a final fiscal cliff deal.

Under Senate rules, there’s no way to block calling up what’s known as a “message from the House” — which is effectively a House amendment to a bill previously passed by the House and amended by the Senate. In effect, Boehner’s proposal could allow Reid, D-Nev., to avoid a filibuster of a motion to proceed.

A senior Senate Democratic aide acknowledged that, with now only six days left before Christmas as of Wednesday morning, Reid could look for every procedural assist possible if a last-minute agreement needs to start its journey to President Barack Obama’s desk on the floor of the Senate.

However, the aide noted the real trouble in either case could be objection to ending Senate floor debate on the measure rather than on starting the debate, which could take three days to overcome.

Using the Myanmar sanctions vehicle has another advantage in the House. There is no opportunity for the House Democrats to offer their own amendment in the form of a motion to recommit the measure.

For the GOP, the measure that Boehner on Tuesday termed his “plan B” should talks with the White House break down also is part of a voting track the speaker is using to counter objections from House conservatives that it asks them in effect to vote for a tax increase. Boehner’s office emphasized Monday the speaker’s favored amendment “does not raise taxes. It is a net tax cut that prevents a $4.6 trillion tax hike on Jan. 1.” That rhetorical point is built in part on the current law that has taxes jumping at the start of the New Year.

His favored amendment would let income tax rates increase only for family income above $1 million, while the other would extend current rates for married couples filing joint returns earning up to $250,000.

Both amendments include adjustments to provide relief to keep the alternative minimum tax (AMT), which was never indexed for inflation, from applying to middle-class families, although the $1 million amendment that may be more widely favored by Boehner and other Republicans would actually eliminate the AMT rather than provide another patch.

Neither plan addresses other looming issues, including the automatic spending cuts under sequestration or the cut in payments to medical providers for treating patients under Medicare. The “doc fix” and the patch to block the expanded AMT hit are often combined on Capitol Hill in what has become a rather ritualistic can-kicking exercise.

Alan K. Ota contributed to this story.