Two easy wins now in doubt because there is not enough time.
Allies, including Republican members of Congress, have long advised Trump that firing Sessions - especially before the November midterm elections - would be deeply damaging to the party.
Flake, a frequent critic of the president, was condemning Trump's tweets earlier Monday in which he attacked Sessions for pursuing criminal charges against GOP Reps.
But he was apparently referring to the first two Republicans to endorse him in the GOP presidential primaries.
Trump wrote on Twitter the Justice Department's decision to file charges will hurt safe Republican seats in the U.S. House of Representatives.
He added sarcastically: "Good job Jeff ..." "Instead of commenting on ongoing investigations and prosecutions, the job of the President of the United States is to defend the Constitution and protect the impartial administration of justice".
Mr Trump then compared the Attorney General with "Lyin' James Comey", claiming the Democrats must love Mr Sessions now.
While discussing the various legal problems Trump faces, not least of which are related to his anger at Sessions for recusing himself from the Russian Federation investigation that led to special counsel Robert Mueller's appointment, the host asked Washington Post bureau chief Phil Rucker about the state of the president's legal team.
In a second tweet, Trump took aim at Democrats, and compared Sessions to fired Federal Bureau of Investigation director James Comey: "Really sick!" Immediately he became a wonderful man, a saint like figure in fact. The president did not elaborate on whether he would keep Sessions on as attorney general following the election. Last month, the president tweeted that Sessions "doesn't understand what is happening underneath his command position" and told "Fox & Friends" that his attorney general had "never taken control of his department".
Justice Department spokeswoman Sarah Isgur Flores declined to comment on Trump's tweets, which did not name the congressmen.
The tension between Trump and Sessions boiled over recently with Sessions punching back, saying that he and his department "will not be improperly influenced by political considerations".
Mr Trump suggested Mr Sessions could lose his job in July a year ago when he said "time will tell" what happens to him.