President Trump, a supporter of Bryant, backs the plan, according to a person familiar with the situation, though there are several other options that McConnell and Trump have discussed if Bryant declines. Like others interviewed for this article, the person spoke on the condition of anonymity to describe private talks.
McConnell is determined not only to protect his majority but add to it in November's midterm elections, and the behind-the-scenes maneuvering over Mississippi reflects his attempt to avoid the calamity of the Alabama Senate race. Last year, conservative firebrand Roy Moore stunned establishment Sen. Luther Strange in the Alabama primary runoff, then lost the longtime GOP-held Senate seat to Democrat Doug Jones.
Cochran, 80, who missed several weeks last fall while recuperating from a urinary tract infection, has appeared frail since his return to the Senate. Elected in 1978, he has not announced any plans for his future. His top aide said Thursday that no imminent plans have been made to announce his retirement.
But Senate leaders and White House aides have been preparing for the possibility that he will step down in the coming months.
In particular, a self-appointment by Bryant would complicate the ambitions of state senator and attorney Chris McDaniel, a foe of McConnell, who ran a failed primary campaign against Cochran in 2014, and has been preparing for another run.
McDaniel previously approached aides to Bryant asking to be considered for an appointment to the Cochran seat, and was told that such an appointment was unlikely, according to people familiar with the situation. McDaniel also has considered running against Sen. Roger Wicker (R-Miss.), who is up for reelection this year. The filing deadline for that race is March 1.
A spokesman for Bryant, who will leave office because of term limits after next year, did not respond to questions about McConnell's request. McConnell spokesman Don Stewart said, "I don't have any readouts of conversations."
"Mississippi is stronger because of Sen. Cochran's service, and I look forward to it continuing," Bryant said in an emailed statement to The Washington Post. "Speculation about anything else is insensitive, irresponsible and unfair."
McConnell and Bryant dined together Tuesday before the State of the Union, which Bryant attended as McConnell's guest. The odds that Bryant will decide to take McConnell up on the suggestion are slim, said a Mississippi Republican strategist familiar with the governor's thinking.
Cochran's chief of staff, Brad White, said that his office helped Bryant get a ticket to the State of the Union from McConnell.
"Governor Bryant asked for Senator Cochran's help in securing a ticket to the State of the Union," White said in an email. "Since Senator Cochran had already given his ticket out, we reached out to the Leader's office for help in meeting the Governor's request."
White added, "If Governor Bryant and Leader McConnell had any conversations, they'd be the ones who could comment on it."
He said "no plans have been made" for Cochran to announce his retirement as soon as early next month.
The idea of Bryant taking the Senate seat is attractive to Washington Republicans who are looking for a strong candidate who would be certain to win in the heavily GOP state.
McDaniel told The Washington Post late last year that he would announce in January whether he would run against Wicker. In an interview Thursday, McDaniel said he postponed his decision "because of the multiple options" before him.
"It's a really fortunate time for me and I'm incredibly blessed to have these options," he said.
McDaniel said he has heard "rumors" that McConnell has encouraged Bryant to appoint himself. "McConnell's doing his best to find someone he thinks can hold off a strong challenger," he said. "But, I just think the people want to have their say."
Allies of Wicker have grown increasingly confident in recent weeks that McDaniel will not challenge him. While they caution that nothing is 100 percent until the March 1 filing deadline, they believe he has increasingly turned his eye toward Cochran's seat.
McDaniel said pursuing Cochran's seat is an "attractive possibility," assuming that changes are made in his seat. He said he wants Cochran to "be healthy and happy."
Two major Republican donors, the hedge fund investor Robert Mercer and the Illinois businessman Richard Uihlein, have given more than $1 million to an independent super PAC, Remember Mississippi, which has been set up to support a prospective McDaniel campaign this year.
"We are certainly hoping that Chris jumps into the Senate race. Which race, we don't know," said Kristina Hernandez, a spokeswoman for the group.
Under Mississippi election law, a resignation by Cochran in the coming months would trigger a special election for the seat to be held on Nov. 6. Unlike the regular election for Wicker's seat, candidates for the Cochran seat would not compete in a primary, and if no candidate received more than 50 percent of the vote in November, the top two finishers would face off in a runoff.
That could provide an easier route for McDaniel to win a seat than running directly against Wicker, a popular incumbent who has the support of Trump.