MILWAUKEE - The behind-the-scenes jockeying for who would replace Sen. JD Vance, R-Ohio, in the Senate has already begun just hours after former President Trump nominated the first-term lawmaker as his running mate.

Names already being floated include Rep. Mike Carey, R-Ohio, former Republican presidential candidate Vivek Ramaswamy, and Ohio Lt. Gov. Jon Husted, Fox News Digital is told. The choice will be up to Ohio Gov. Mike DeWine, who would appoint someone to finish Vance's term.

Carey, who has strong relationships with both Trump and DeWine, reached out to DeWine shortly after Vance was selected, a source familiar with the discussion said. They also met last week on a separate matter.

That source pointed out that Carey had a voting record that lines up with DeWine's principles but was also endorsed by Trump for re-election.

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JD Vance is introduced during the Republican National Convention

Former President Trump picked Ohio Sen. JD Vance to be his running mate on Monday (Win McNamee/Getty Images)

A spokesperson for Carey said in response to the speculation, "From his decade serving in the military to the day he was sworn into office, Congressman Carey has been dedicated to service. He remains focused on representing Ohio's 15th district in Congress."

DeWine, widely regarded as part of the establishment wing of the party, is likely to face pressure from Trump loyalists to appoint a candidate in line with Trump's agenda and former presidential candidate Vivek Ramswamy has been floated as a possibility.

Ramaswamy indicated he was interested in the potential vacancy when reached by Fox News Digital.

"I have rock-solid conviction that JD will be an outstanding VP for President Trump. The only negative is this leaves our side missing one of our best fighters in the Senate. If asked to serve, I would strongly consider the position and would discuss with President Trump which path makes the most sense for our country," Ramaswamy said.

Ohio GOP State Sen. Matt Dolan, who was endorsed by DeWine in the Senate race against Bernie Moreno earlier this year, is also believed to be in the running to replace Vance, a GOP Senate campaign strategist told Fox News Radio.

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Mike Carey with hands folded

Rep. Mike Carey, R-Ohio, is one of the names floated as a possible replacement if Trump and Vance win.

Dolan congratulated Vance in a social media post shortly after Trump's announcement and said the decision is "great news for Ohio and America."

DeWine would not give any indication of his thinking when Fox News Digital pointed out he had a choice to make.

"We have to win first," DeWine told Fox News Digital on the floor of the Republican National Convention (RNC).

In a statement to Fox News Digital, DeWine Press Secretary Dan Tierney said, "I do not anticipate the Governor will comment on the appointment process until there is an actual vacancy after Election Day."

"The Governor has said his focus through November will be on helping elect this ticket, our Senate candidate Bernie Moreno, and the Republican slate of Ohio Supreme Court candidates."

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 Vivek Ramaswamy pointing while speaking

Former Republican presidential candidate Vivek Ramaswamy also indicated he was interested.  (AP Photo/Gerald Herbert)

Husted, who delivered a speech at the RNC before Vance took the stage as Trump's running mate, is also a possible pick for DeWine.

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The state's attorney general, Dave Yost, told Fox News Digital in an interview that he believes Husted would be an "excellent choice."

Another possible contender could be Rep. Warren Davidson, R-Ohio, who was briefly considered as a possible challenger to Sen. Sherrod Brown, D-Ohio, in this year's elections before announcing he would instead seek another term in the House.

Fox News Digital reached out to Husted, Carey, and Davidson for further comment.