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Chuck Grassley.
Grassley, an Iowa Republican, first made the comments in a Wednesday night tweet. He followed up on them when asked by CNN's Manu Raju Thursday.
"My agenda is full," he told Raju, noting that his committee won't act on any successor to Sessions, who finds himself under fire from the president.
"Everybody in D.C. Shld b warned that the agenda for the judiciary Comm is set for rest of 2017," Grassley tweeted Wednesday night. "Judges first subcabinet 2nd / AG no way."
Concerns about a possible replacement for Sessions have peaked over the past week as Trump has continued to publicly express his frustration with the attorney general.
Trump is displeased with Sessions' recusal from all investigations involving the Trump campaign, which allowed for Deputy Attorney General Rod Rosenstein to appoint special counsel Robert Mueller to oversee an investigation into whether members of the Trump campaign colluded with Russian officials in the 2016 election. Rosenstein appointed Mueller after Trump abruptly fired James Comey as FBI director in May. Comey was overseeing the FBI's investigation into Russian election interference.
On Wednesday, Trump fired off a pair of tweets asking why Sessions has not "replaced" acting FBI Director Andrew McCabe. And the president started his day on Tuesday by blasting Sessions on Twitter for having taken "a VERY weak position on Hillary Clinton crimes (where are E-mails & DNC server) & Intel leakers!" The president also asked "where is the investigation A.G." in a tweet in which he discussed "Ukrainian efforts to sabotage Trump campaign."
In an interview with The Wall Street Journal later that day, Trump said he was "very disappointed in Jeff Sessions," a statement he would repeat in a Rose Garden press conference alongside Lebanese Prime Minister Saad Hariri just hours later.
Trump has not, however, said publicly whether he plans to oust Sessions.
But as The Washington Post reported Wednesday night, the president has discussed the possibility of pushing Sessions out and appointing a new attorney general during the Senate's August recess. The Post reported, however, that Trump was warned not to make such a move because of the likely fallout.
The idea of a recess appointment gained traction as Trump's attacks on Sessions continued, but Democrats and Republicans alike were quick to shoot down the notion of such an appointment, saying they would not allow it to occur.
Normally, a potential Sessions replacement would have to face Senate confirmation, a process that would be extremely difficult for any potential nominee to pass even as the Senate is under Republican control. Based on Trump's public comments, it would look as though Trump replaced Sessions solely for the purpose of selecting an attorney general who would be willing to fire Mueller.
If Trump were able to make an appointment while the Senate is in recess, that person could serve until the end of the next Senate session in early 2019 without being confirmed.
Already, however, senators are "planning to continue procedural moves" that would prevent the Senate from "formally adjourning for recess," seeking to prevent Trump from making any sort of recess appointment, CNN reported Tuesday.
A number of Republicans also shut down the idea of a recess appointment on Thursday.
Speaking from the Senate floor, Sen. Ben Sasse of Nebraska, a Judiciary Committee member, said "if you're thinking of making a recess appointment to push out the attorney general - forget about it."
"The presidency isn't a bull, and this country isn't a china shop," Sasse continued. "Mr. President, you're a public servant in a system of limited government with a duty to uphold and to defend and to teach to our kids the Constitution system of checks and balances. And this, this is the world's greatest experiment in self-government. It works only if all of us, presidents, senators, Republicans, Democrats, independents, and judges - if we all keep our faith to the American institutions and to the rule of law."
A Republican senator told The Washington Examiner that Senate Republicans are working to ensure that Trump cannot make any recess appointments.
"Recess appointments have never been an option for Trump," said the Republican senator.
Elsewhere, Republican Sen. Lindsey Graham of South Carolina told CNN that "there will be holy hell to pay" if Trump ousts Sessions, adding it could be "the beginning of the end" of Trump's presidency if he tries to go after Mueller.
Also a Judiciary Committee member, Graham added "there will be no confirmation hearing for a new attorney general in 2017."