If Mr. Trump wants to unilaterally install anyone else as attorney general without going through Senate confirmation, he might be able to do so under the Constitution’s recess appointment clause. It empowers presidents to fill vacant positions during Senate recesses in excess of 10 days. A recess-appointed attorney general would stay in power until January 2019, when the next Congress is seated.

Will the Senate’s August recess present an opportunity?

It could. If the chamber goes into a formal recess before leaving Washington next month for its annual break, Mr. Trump could make recess appointments while senators are on vacation.

Can the Senate block Trump’s recess appointments?

Yes. Congress can prevent presidents from making appointments during its breaks by holding so-called pro forma sessions, in which a senator comes into the otherwise-empty chamber every few days and bangs the gavel. That technically breaks up a long recess into a series of short ones — each too brief to trigger appointment powers.

Could Democrats alone block Trump?

That is not clear. The Senate minority leader, Chuck Schumer, Democrat of New York, called upon Republicans on Tuesday to use pro forma sessions in August to prevent Mr. Trump from appointing “someone during the August recess who will fire Special Counsel Mueller and shut down the Russia investigation.” He also vowed that Democrats would try to do that on their own if Republicans refused.

Before the Senate adjourns for more than three days, Congress adopts a concurrent resolution giving it the authority to do so. That is normally routine, but Democrats are discussing whether to try to block any up-or-down vote on such a resolution by offering an unlimited series of amendments to its date and time for reconvening. If the Senate gives up and leaves town without such a resolution, the Constitution would require pro forma sessions every three days, blocking recess appointments.

Could Republicans overcome such a move?

Alan S. Frumin, who served as the Senate’s chief parliamentarian from 1987 to 1995 and from 2001 to 2012, said Republicans could “quash such a filibuster by amendment” using what some call the nuclear option to change the rules. Under that scenario, if a majority of senators vote to deem the Democrats’ amendments “dilatory,” they could adjourn in a way that leaves Mr. Trump free to make recess appointments. But if at least three Republican senators break ranks, the Senate would need to meet every three days.

The Constitution also says that if the House and Senate disagree when to adjourn, the president “may adjourn them to such time as he shall think proper.” That suggests Mr. Trump might be able to force the Senate into a lengthy recess. But no president has tried to exercise this power, and it is not clear what would constitute such a disagreement: Mr. Frumin said that the House typically gives its consent for the Senate to adjourn for more than three days but does not order the Senate to do so.