However, some experts have said the offer made by Trump in Texas on Wednesday could be seen as a pledge to cut taxes, which is legal.
Dr Joan Mahoney, lecturer in law at the University of Southampton, told the BBC she thought the offer would be legal as it would be similar to a campaign promise - though she said politicians "don't usually do it quite this explicitly".
"On the one hand it sounds a bit like a bribe, vote for the Republicans and you'll get money, which would be decidedly illegal," she said.
"But on the other hand it would go to everyone, regardless of how they voted so it isn't in that sense a bribe.
"I think it would be legal. It's the same kind of campaign pledge as saying 'if my party wins we'll reduce taxes'. Every party does that all the time... and that's obviously perfectly legal. "
However, Laurel Rapp, director of the US and North America programme at Chatham House, told the BBC that Trump's offer seemed to be "a very different, very transactional agreement" from the kind of promise usually made by candidates in an election period.
There are nearly 270 million American adults, meaning a $5,000 payout would cost the US government some $1.3 trillion.
Trump gave no details on how such cheques would be funded during his speech.
Congress authorises spending in the US, but the president did not say whether he planned to ask for legislative approval for any payments. And it seems highly unlikely he could approve such a vast sum via presidential executive order.
Erica York, a senior economist at the Washington DC-based Tax Foundation, told the BBC she would be "extremely surprised" if Congress approved any proposal.
"This would be one of the worst ways to use $1.3 trillion," she said.
"We're on a very unsustainable track, with deficits only projected to grow and grow," York added.
Speaking to Fox News after Trump's speech, Vance indicated that the payments could be funded by money raised by the president from tariffs.
However, such revenue to date falls far short of the money that would be needed to account for Trump's $5,000 cheques offer.
The US president also told the Republican convention that the money would have to be spent in the US, but it is not clear how that would be monitored.
According to the Bipartisan Policy Center, the US has brought in around $475bn (£203bn) in tariff and excise tax revenue since the beginning of 2025.
However, more than $100bn (£74bn) has been refunded after many tariffs earlier this year.
When asked how the payouts could work, the White House said in a statement to the BBC: "The doomers and naysayers have consistently doubted President Trump. "
"With the continued support of the American people, President Trump will keep delivering real results," added spokesman Davis Ingle.