Via the Right Scoop, Reid floated this deadline on Friday too. Dick Durbin, another member of the leadership, has also been talking it up. The clock starts on June 10th, when most Republican primaries will be over and the party will once again be free to betray its conservative supporters, and ends when Congress leaves town for the August recess. If Boehner hasn’t passed a bill by then, Obama’s going to do something on deportations — probably something marginal and gestural, just to show lefties that he cares, but one never knows with a guy who’s gotten as comfortable as O has with executive action.

Do you suppose these endless Democratic threats about an impending moratorium on deportations might be influencing the rate of illegal immigration these days? Let’s check in at the Texas border:

A shift by immigrant smugglers to target the tough-to-defend U.S.-Mexico border in southernmost Texas has accelerated this year as the Border Patrol scrambles to shift its resources from states farther west, according to an internal agency report obtained by The Associated Press. From Oct. 1 through May 17, agents in the southernmost tip of Texas made more than 148,000 arrests, on pace to match last year’s total in less than eight months, according to the intelligence report. That compares with nearly 63,000 arrests in the Tucson, Arizona, sector, which it surpassed for the first time just last year. The Rio Grande Valley sector averaged nearly 1,100 arrests per day from May 11-17, according to the document. What these numbers look like on the ground is a near-constant flow of people across the Rio Grande.

It’s not just Texas. The Border Patrol started seeing higher rates of border-crossing last year, when the Senate first got cracking on immigration reform. That’s a feature, not a bug, of lefty table-pounding about this. The more amnesty dominates the policy agenda in the U.S., the more of a rush there’ll be by would-be illegals to sneak in under the wire before any new policies, whether legislative or executive, take effect.

Now, a strategic question for Boehner: Does he call Obama’s bluff by walking away from reform this summer or does he pass something small, like ENLIST? The fear if he walks away is that Obama will do something dramatic, Latino voters will rally for Democrats, and suddenly not only have the midterms gotten more difficult but any amnesty bills passed by the GOP next year will seem like an afterthought to O’s action. Chances are, though, that O’s move on deportations will be modest so as not to spark a backlash among white voters and that Latinos will largely ignore it (or even be irritated by how modest it is). In that case, if you’re Boehner, why not walk away for now and focus on the lame-duck session? Republican voters will be grateful on election day and then you can betray them later. On the other hand, passing ENLIST has its own advantages. It’s big enough that the party could sell it to Latinos as proof of its goodwill but small enough that it would amnestize only a tiny sliver of the illegal population. It would give Democrats fits, too, because they don’t want to hand the GOP a political victory unless they’re getting a lot of illegals legalized in return — but they also don’t want to be seen as opposing a bill that represents a rare Republican gesture towards amnesty. The risk to Boehner, though, is that conservatives might rally against ENLIST and that might hurt the party, both in angering some voters into staying home this fall and also “proving” to immigration fans that there’s no form of legalization that conservatives will support.

My hunch is that they’re going to push ENLIST tentatively in June and see what the reaction is, both inside Congress and out. If Boehner can bring dozens of Republicans onboard (if not a clear majority) and grassroots righties seem no more than mildly annoyed, he’ll go for it and then dare Reid and Schumer to reject the bill on grounds that it doesn’t amnestize enough people. If Democrats walk away, he can walk away too saying, “I tried.” No harm done for the midterms. Or at least, not much.