The Congress, whenever two thirds of both Houses shall deem it necessary, shall propose Amendments to this Constitution, or, on the Application of the Legislatures of two thirds of the several States, shall call a Convention for proposing Amendments, which, in either Case, shall be valid to all Intents and Purposes, as Part of this Constitution, when ratified by the Legislatures of three fourths of the several States, or by Conventions in three fourths thereof, as the one or the other Mode of Ratification may be proposed by the Congress; Provided that no Amendment which may be made prior to the Year One thousand eight hundred and eight shall in any Manner affect the first and fourth Clauses in the Ninth Section of the first Article; and that no State, without its Consent, shall be deprived of its equal Suffrage in the Senate.
All Amendments so far have been achieved by the first process, in which it is proposed to the states by a two-thirds vote in both the House and the Senate and none by the second, in which the states propose them to themselves. However, Michigan recently may have triggered the process by becoming the thirty-fourth state to call for Constitutional convention. The tricky thing is how, exactly, one counts the two-thirds of the states. Thirty-four states have specifically called for a Constitutional convention on the matter now. However, roughly about a third of the states rescinded their resolutions for it. So one might think that they don't count. On the other hand, Article V only requires that the states apply for a Constitutional convention. Are they still counted as applying for one if they actually apply but then say that they take it back? Also, there are likely various differences in what states wanted a convention called for; does the mere calling for a convention suffice, or do the states have to agree on why? There are no established precedents. The closest we have come to one is with the Equal Rights Amendment, when the courts established in Idaho v. Freeman that, when an Amendment has been proposed, states have full right to rescind their ratifications as long as it is prior to its having achieved the two-thirds threshold. But in part that was a matter of deference to the states in a matter with no consequences; the real question here would be which way Congress moves. That, of course, will be determined on grounds of political expedience. When the Twenty-Seventh Amendment passed, there were ambiguities, and Congress simply recognized the Amendment, despite being blindsided with it and despite the fact that they didn't like it, for the obvious reason that they were all sufficiently intelligent to realize that Congress refusing to recognize an Amendment restricting their ability to give themselves pay raises would create a political storm they could not survive.
It's a little less clear what would happen here; balanced budget amendments are extremely popular with voters, but Congress has considerable room to maneuver politically. The most rational first move would be simply for Congress to make a formal request of the states to confirm their applications so that it can determine whether it is supposed to call a convention. It's a situation in which Congress has plenty of incentive to proceed at a leisurely pace. My suspicion is that it will require 34 clear and unambiguous cases, unless they get worried that it might be an election issue. It's unlikely to do so, but it may even then encourage Congress to propose a balanced budget Amendment on their own terms; that's what happened with the Seventeenth Amendment: the Senate fiercely opposed any Amendment to change how they were elected until it began to look likely that the reformers could force the convention route.
In any case, the House of Representatives has been formally asked to investigate and determine whether and how to proceed.