Washington (CNN) Former Attorney General Jeff Sessions might be interested in jumping into the race for his old Senate seat, but some Republicans are throwing cold water on the idea.

Sessions, who represented Alabama in the US Senate from 1997 until he became President Donald Trump's first attorney general in 2017, is giving the race a look, a source familiar with the matter confirmed to CNN on Monday. He would need to make a decision by November 8.

But Sessions has not discussed the race with Senate Majority Leader Mitch McConnell, nor has McConnell encouraged Sessions to run, according to one GOP source familiar with McConnell's thinking. McConnell, a Kentucky Republican, believes "Republicans are very well positioned to retake the seat" as it is, the source added.

Even though controversial former Judge Roy Moore is running once again , Republicans do not believe he is a serious threat in the primary, according to another Republican source who is following the race. The GOP is confident they have a strong candidate at this stage in former Auburn University football coach Tommy Tuberville, the source said.

It's also unclear how Sessions would fare in light of Trump's past attacks on his former attorney general for his handling of the Mueller investigation . Trump remains extremely popular in Alabama, and he "will 100% go after Jeff Sessions" if Sessions runs, the second GOP source predicted.

Alabama Secretary of State John Merrill, who is one of the GOP candidates vying for the seat, told CNN that Sessions "immediately becomes the front-runner" if he gets into the race. But Merrill acknowledged that Sessions' chances could be complicated by his relationship with Trump, which was badly strained after Sessions recused himself from the investigation into Russia's meddling in the 2016 election, leading to the appointment of a special counsel.

"Sen. Sessions has made himself vulnerable with some of the things that have happened" with the President, Merrill said. "That would be something people who support Sen. Sessions would have to give consideration to."

"If the President were to come out publicly against Sen. Sessions," Merrill added, "I think it would be very difficult for Sen. Sessions to overcome that."

Merrill would not say whether he would end his Senate bid if Sessions were to run. But, Merrill said, "I can tell you I have always been a fan of Sen. Sessions and always been a strong supporter."

And Tuberville and Merrill aren't the only other Republicans in the primary race. US Rep. Bradley Byrne, who is also running, told CNN that he'll stay in the GOP primary race even if Sessions enters it. "It's not going to make any difference to me," he said.

"No matter who's in the race at 5:01 p.m. on November 8, I'm in it," said Byrne, referring to the state's qualifying deadline. "I'm in it to win the race. And I'm going to win the race."

When asked about the speculation surrounding Sessions, Byrne responded, "It doesn't matter to me. I'm in."

Byrne said that he and Sessions talked on the phone this month about the Senate race but wouldn't go into further details.

"I'm going to keep doing my job," said Byrne. "I'm going to keep pushing back against this effort to impeach the President."

When asked if he had talked to President Donald Trump about a potential endorsement, Byrne said, "no comment."

Senate Majority Whip John Thune said Tuesday he thinks Trump and Sessions would "look past some of their differences" if it meant Sessions could win back his Senate seat and help Republicans maintain control of the Senate.

"I think in the end it's going to be dependent on winning that seat, and I think the President will be very interested in having a Republican majority in the Senate, and if Sen. Sessions looks like the person that can move that seat into our column, then I'm guessing they will probably look past some of their differences," Thune, a Republican from South Dakota, told reporters on Capitol Hill.

The seat is currently held by Democrat Doug Jones, who won the special election in 2017 after Sessions vacated the seat to join the Trump administration.

Thune, the No. 2 Republican in the Senate, praised Sessions as "someone who I think can win in Alabama."

"He obviously has a lot of experience and seniority he brings to the campaign, and then if he's successful can add to the Senate," Thune said.

This story has been updated with additional developments Tuesday.