Donald Trump Donald John TrumpObama calls on Senate not to fill Ginsburg's vacancy until after election Planned Parenthood: 'The fate of our rights' depends on Ginsburg replacement Progressive group to spend M in ad campaign on Supreme Court vacancy MORE’s lone Senate supporter threw cold water on the notion that he might be chosen as the Republican presidential hopeful's running mate on Thursday.

Sen. Jeff Sessions Jefferson (Jeff) Beauregard SessionsTrump's policies on refugees are as simple as ABCs Ocasio-Cortez, Velázquez call for convention to decide Puerto Rico status White House officials voted by show of hands on 2018 family separations: report MORE (R-Ala.) told reporters outside the Senate chamber that he has no expectation of being Trump’s vice presidential candidate and has not spoken with him about who could be on the ticket.

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“I think that would not happen,” Sessions said. “I have not talked with him about it.”

“Don’t bet any money on me.”

Trump has claimed in recent interviews that he would likely pick an established politician as a running mate if he were to become the GOP presidential nominee.

Trump would choose “somebody that can walk into the Senate and who’s been friendly with these guys for 25 years, and people for 25 years, and can get things done,” he told The Washington Post in an interview published over the weekend. “So I would 95 percent see myself picking a political person as opposed to somebody from the outside.”

The comments raised speculation that Sessions — the only current member of the Senate to have endorsed Trump’s presidential bid — might be on the short list.

The betting website Paddy Power is offering 9-to-1 odds that Sessions will be the Republican vice presidential candidate; that’s better than the 10-to-1 odds for Sen. Marco Rubio Marco Antonio RubioFlorida senators pushing to keep Daylight Savings Time during pandemic Hillicon Valley: DOJ indicts Chinese, Malaysian hackers accused of targeting over 100 organizations | GOP senators raise concerns over Oracle-TikTok deal | QAnon awareness jumps in new poll Intelligence chief says Congress will get some in-person election security briefings MORE (R-Fla.), a former presidential candidate.

Trump has faced a tremendous amount of opposition from within his own party, raising questions about whether he will be able to secure the 1,273 delegates necessary to be the nominee before the summer convention.

In recent weeks, he has been in contact with prominent Republican lawmakers including Sens. Orrin Hatch Orrin Grant HatchBottom line Bottom line Senate GOP divided over whether they'd fill Supreme Court vacancy MORE (Utah) and Tom Cotton Tom Bryant CottonRenewed focus on Trump's Supreme Court list after Ginsburg's death Republicans call for DOJ to prosecute Netflix executives for releasing 'Cuties' Loeffler calls for hearing in wake of Netflix's 'Cuties' MORE (Ark.), which some have interpreted as an attempt to unify the party.

Hatch on Thursday told reporters that he talked by phone with Trump last month, and that it had been a "very decent conversation.”

“I think highly of him, there's no question about it,” Hatch said.

In coming weeks, that type of outreach appears likely to intensify. It has been reported that Trump's campaign will begin regular meetings with lawmakers on the Hill, starting as early as next week.

“I think we’ll be able to see a strengthened Washington presence, which I support,” Sessions said. “And I think we’re going to see a slowdown in the rapidity of these primaries, a better opportunity for him to begin to talk personally with more key people. “

Naomi Jagoda contributed