Donald Trump has reportedly dropped the biotech entrepreneur and former presidential hopeful Vivek Ramaswamy from his list of potential vice-presidential picks, considering him for a cabinet role instead.

Bloomberg News cited “people briefed on the discussion” in reporting that Trump “personally told Ramaswamy he won’t be his vice-presidential pick … but is considering him for posts including homeland security secretary”.

Ramaswamy is seen as suited for that job, Bloomberg said, because he is the son of immigrants from India, which might help “neutralise” criticism of hardline immigration measures.

Promising to put migrants in camps and institute mass deportations, Trump has repeatedly said migrants are “poisoning the blood of our country” – rhetoric widely compared to that of Adolf Hitler.

Representatives for Ramaswamy did not immediately respond to a request for comment.

Trump is the presumptive Republican nominee despite facing 88 criminal charges (for election subversion, retention of classified information and hush-money payments) and multimillion-dollar civil penalties his lawyers say he is struggling to pay.

Trump’s former vice-president, Mike Pence – whose time in the role ended with Trump supporters chanting for his hanging on January 6 – said last week he would not endorse Trump this time. He did not say he would not vote for him.

Jason Miller, a Trump adviser, told Bloomberg speculation was irrelevant.

“Apparently somebody has decided to list out everyone who has ever met President Trump and is now speculating as to their potential participation in a second Trump administration,” Miller said. “The truth is that unless you hear it directly from President Trump or his campaign, this is all BS.”

That did not stop Bloomberg reporting that Elise Stefanik of New York, the No 3 US House Republican, has dropped off the vice-presidential list but is in line for a cabinet post should Trump defeat Joe Biden.

Among other possible cabinet picks, Bloomberg named Doug Burgum, the North Dakota governor who ran for the Republican nomination, and Robert Lighthizer, US trade representative in Trump’s first administration.

Bloomberg also reported that Kevin McCarthy, who Trump supporters made the first House speaker ejected by his own party, was a “top candidate” for chief of staff.

Two far-right senators, Mike Lee of Utah and Ted Cruz of Texas, are reportedly seen as possible picks for attorney general.

Tim Scott, the South Carolina senator who ran for the nomination but switched to declaring his “love” for Trump, has reportedly been pushed as a vice-presidential pick by allies including Lindsey Graham and John Thune, both senior Republican senators.

Trump sounded less than enthusiastic about Scott when talking to Sean Hannity last month. “Tim, for himself, he was fine,” Trump said. “He did OK. I mean, he was OK as a candidate, but he didn’t want to talk about himself. He’s a very good man. For me, he’s unbelievable. He’s a surrogate.”

“Nothing is decided,” Bloomberg concluded, “and it will likely be a game of musical chairs for top posts.”

One figure perhaps unlikely to find a seat when the music stops is Couy Griffin, the Cowboys for Trump founder who was removed from office under the 14th amendment to the US constitution thanks to his participation in the January 6 insurrection.

On Monday, after the US supreme court declined to consider his removal as commissioner of Otero county, New Mexico, Griffin pointed to the court’s rejection of an attempt to remove Trump from the ballot under the same constitutional measure.

“As I sit right now,” Griffin said, “the only office I can run for is the executive office. Trump needs a vice-president who can stand strong through the hardest of times. And I can only pray I’d be considered.”

Griffin, who was sentenced to 14 days in jail and fined $3,000 for misdemeanor trespassing during the Capitol attack, also tweeted a picture of himself with Trump in the Oval Office.

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