If the bank gets extended only into early next year, its charter expiration could then coincide with the need to raise the debt ceiling and again make it an attractive bargaining chip in deliberations to pass a bill avoiding catastrophic default.
Read MoreWhat's behind the Ex-Im Bank battle
But if GOP opponents of the bank can extend it just past the raft of other deadlines, the internal party thinking goes, they can simply do nothing in June and watch Ex-Im disappear, giving party activists a big win and not risking major political blowback.
The Republican case for killing Ex-Im in June will gain even more strength if the party takes the Senate in November, which still seems like a probability, though not an overwhelming one given the strengthening economy and stronger than expected Democratic performance in some swing states.
The House GOP will also likely have a somewhat stronger majority next year, further enhancing the likelihood that those in the party who support Ex-Im, a group that includes more moderate Chamber-of-Commerce-type Republicans, will have less sway.
The only potential risk to the House GOP strategy of extending Ex-Im to June and then letting it die is that it gives supporters more time to marshal their case for a full charter extension. And there are many deep-pocketed and politically powerful supporters of Ex-Im, including the chamber, the National Association of Manufacturers and other big business groups.
Read MoreBoehner signals temporary extension for Ex-Im bank
Those groups plan to continue to highlight the extent to which Ex-Im guarantees to foreign buyers make it possible for small businesses to export products and hire more U.S. workers. Every time a GOP member of Congress gets a call or visit from a constituent saying he or she will have to lay off workers if Ex-Im goes away makes it that much harder for the member to allow the bank to vanish.
In addition, pushing Ex-Im off to June takes it out of a heated political environment.
Some Republicans who otherwise might support the bank don't want to do so closer to an election where they could face tea party retribution. But June is far enough away from an election that members could both please big business backers and avoid major criticism by supporting an Ex-Im reauthorization.
But none of that could even come into play because all the House GOP leadership would have to do in June is nothing. They could simply not bring Ex-Im up for a vote and allow its charter to expire and see it go out of business. For that reason, the bank and its supporters better hope Senate Democrats kill the June extension next week.
—By Ben White. White is Politico's chief economic correspondent and a CNBC contributor. He also authors the daily tip sheet Politico Morning Money [politico.com/morningmoney]. Follow him on Twitter @morningmoneyben.