But doing it after the Midterms solves only a political problem. It does nothing to temper the potentially game-changing effect on the Russia and related investigations.
And that's the big question that is likely to remain, given that Robert Mueller's and the Southern District of New York's Michael Cohen cases aren't done. Trump clearly wants an attorney-general who will oversee these probes - unlike Sessions, who recused himself from Russia-related matters - and his actions suggest he wants someone who will do his bidding. Replacing Sessions could, theoretically at least, give him both.
But it's not quite so simple. And the whole thing could quickly devolve into a big political and legal battle.
There would be two main dominoes to fall after Sessions resigns or is removed:
1) Trump may pick an acting attorney general, and ...
2) He would nominate a full-time replacement, who would be subject to confirmation.
Both could conceivably take oversight of Mueller's Russia investigation away from Deputy Attorney-General Rod Rosenstein, and if they're more loyal, logic suggests, they could run interference for Trump or even try to shut Mueller's probe.
Getting to that point, though, will require clearing several hurdles.
The first is whether Trump can even pick an acting AG. As you may recall from David Shulkin's exit as veterans affairs secretary, there is an open question as to whether the Federal Vacancies Reform Act allows a president to temporarily replace a Cabinet official he has fired - as opposed to one who has resigned.
If Sessions forces Trump to fire him, there could be a legal battle over Trump's authority to pick a temporary replacement who would outrank Rosenstein. Otherwise, Rosenstein would effectively take over. So there's no guarantee of success for Trump there.
If Trump were able to pick a temporary replacement, it couldn't be just anybody; it would have to be someone who has already been confirmed by the Senate. That could be someone already at the Justice Department, or it could be a loyalist from outside.
Former Deputy Assistant Attorney-General Marty Lederman said these are probably the biggest questions when it comes to whether Trump can actually affect the course of the investigations he faces.
"Whomever Trump appoints as a so-called acting AG in the short term will probably have greater practical significance than whom he nominates for Senate confirmation," Lederman said.
That's in part because it would be the person who could take over immediately, and in part because it would be someone who wouldn't be subject to political manoeuvering. The full-time replacement, after all, needs to be confirmed.
And while Senate Republicans are warming to the idea of replacing Sessions, that doesn't mean they will give Trump carte blanche. Given the GOP's slim 50-to-49 Senate majority - soon to be 51-49 once the late Senator John McCain's replacement is appointed - Trump would need to strike a balance between installing someone who could do what he wants and someone who can win over Senate Republicans who at least profess to want the opposite.
Senator Ben Sasse, (R), for instance, has said, "I find it really difficult to envision any circumstance where I would vote to confirm a successor to Jeff Sessions if he is fired because he is executing his job rather than choosing to act like a partisan hack."
Senator Susan Collins, (R), has said she doesn't think a new AG will even be confirmed if Trump fires Sessions.
Of course, that's not quite the same as completely ruling out voting for a replacement - circumstances change, as Graham showed us - nor does it address what would happen if Sessions resigns.
If Republican do entertain the nominee, the nominee may be grilled on whether he or she will allow Mueller to finish or even whether the nominee would choose to recuse from the probe, like Sessions did. Of course, acceding to either of those demands would basically defeat the purpose, from Trump's perspective, and may just create the same situation we currently have with Sessions.
"It is highly doubtful that the President will appoint anyone who offered the slightest reservation about taking over for [Rosenstein] in the Mueller investigation," said former federal prosecutor Jeffrey Cramer. "It seems like that is the first criteria for this White House."
Whoever replaces Sessions - either in an acting or full-time capacity - the big questions at that point would be whether that person would undo what Rosenstein has done, restrict Mueller's authorities in some new way, or even retract Mueller's appointment.
Absent those clearly controversial potential moves, it would really be about other new Mueller authorities, charges and immunity deals that may come about. (Mueller doesn't appear to think he can indict a sitting president.)
Another interesting subplot is whether the new attorney general or acting attorney general might also insert themselves in the Cohen probe, in which the former Trump lawyer has just implicated the president in campaign-finance violations over hush-money payments. But SDNY has historically been very independent of the main Justice Department, and that would lead an internal clash, according to experts.
"It wouldn't be cost-free for a new AG to significantly constrain the Mueller or SDNY investigations," Lederman said. "Trying to impose serious limits would, in some cases, raise legal questions and, at the very least, would likely result in significant institutional resistance and tumult."
Ultimately, as with so many other things, this boils down to how much Republicans are willing to put up with from Trump - and whether they feel they can stop him.
Also, as with so many other things, it will probably take only one or two of them to actually stand in his way.
Mostly, they haven't been prepared to actually fight him, or they've fought him only partially, succumbing to the politics of the day in the GOP. This situation could be different for a whole host of reasons, but as Graham demonstrated, we also haven't seen anyone draw a true line in the sand and stick to it yet.
