<sigh> Of course he does.....
....but let's remember that for which he's claiming credit:
First of all, don't believe for a minute that DHS is actually planning a genuine illegal alien deportation offensive. They're only pretending to to.....actually, I can't think of any reason why they would want to create the public impression that they're functionally abrogating their own demigod's imperial amnesty decree from thirteen months ago. Boredom, perhaps? Maybe it's "Opposite Day" at the White House in lieu of Christmas? Beats me.
Whatever the reason, it is guaranteedly not serious. Why, you may be asking? Because they're announcing their planned "raids" a month in advance....
Whether O's make-believe "Deportageddon" is a "political nightmare" for the Empress depends on which phase of the campaign at which you're looking. In the Democrat primaries, it would actually seem to be beneficial to her chances at the Donk nomination, as she would have a foil in her own party off of which to bounce her ham-fisted pandering to La Raza-oids and Reconquistas - "Nominate me and I'll reverse the president's cruel and racist anti-immigrant crackdown!" or something along those lines. To the degree that Democrat voters are some combination of (1) gullible and (2) resigned - a metric at which I'll refrain from speculation, thank you very much - they would presumably eat it right up and snarf it right down. But in the general election, among a general electorate that is still massively opposed to illegal immigration but resigned in despair to being overrun by it, with that issue being one of the hot general campaign topics? Yeah, that could be a major problem for her.
My best guess? The White House believes that Hillary will be the Democrat nominee and is bulldozing her into yet another untenable general election position that The One could overcome with the race card but she won't be able to with her vagina, to say nothing of her abysmal lack of any political skillset.
Trump is taking credit for Obama's immigration equivalent of his phony "war" against the Islamic State, in essence.
Which is....quite fitting, actually: an ersatz "Republican" bragging about an ersatz deportation "offensive". Imagine what he could achieve if he were more than accidentally talented.
(Bazinga - if Trump were actually talented, or intelligent, or substantive, or, you know, not an anti-conservative Putin-lover manipulated into serving as Hillary Clinton's unwitting stalking horse, he'd have been vaporized by the media within twenty four hours of declaring his candidacy and we wouldn't be having this scintillating conversation. But then, if Trump were really a Republican, he wouldn't be such a loudmouthed jackhole, now would he?)
UPDATE: And then there's the question of what possible political pressure Trump's personification of the Dems' insulting caricature of Republicans as "racists" and "xenophobes" the past six months could have put on Obama, who faces no more elections and who illegally, unlawfully, and unconstitutionally imposed amnesty in the teeth of overwhelming public opposition over a year ago. The One has been governing against the will of the people ever since he took office almost seven years ago Even if he had a relevant reason to fear anything The Donald had to say about anything, he wouldn't. But like his self-styled foil, for Trump everything has to be about Trump, the less connected to reality the better.