The South Carolina senator said Mr Trump's hold over the Republican Party and vast swathes of its electoral base could make it difficult for GOP lawmakers on Capitol Hill to back legislation opposed by the outgoing president.
It comes as Mr Trump continues to falsely claim that he won the election and reportedly plans for an "opposition rally" at president-elect Joe Biden's 20 January inauguration.
Despite losing both the popular vote and the electoral college at the 2020 election, Mr Trump secured some 74 million votes - up from 62 million in 2016 and more than other defeated presidential candidate in history.
That increased vote share has the president reportedly mulling over another run for the White House in 2024.
Before then, he could be a thorn in the side of lawmakers hoping to get business done on a bipartisan basis, according to Mr Graham, who believes Mr Trump is favourite for the GOP nomination for president in four years' time.
"He has a lot of sway over the Republican Party," he told The Atlantic in an interview published earlier this week, while still not acknowledging Mr Biden's win.
Mr Graham is among a number of GOP politicians and officials who refuse to acknowledge, at least publicly, that Mr Trump has lost the election.
"If he objects to anything Biden [does], it would be hard to get Republicans on board," Mr Graham went on.
He added: "If he blessed some kind of deal, it would be easier to get something done. In many ways, he’ll be a shadow president.”
The idea of a shadow leader, common in parliamentary systems such as the UK where the opposition party receives public money to set up an office to hold the government to account, is unheard of in the US.
With no office to support them, the defeated presidential candidate usually pursues other endeavours after leaving the White House.
But with a large war chest built up and an army of supporters behind him, Mr Trump may have other ideas.
“Grover Cleveland did it," Mr Graham added, referencing the Democratic president who lost out to Benjamin Harrison but won the next election.
Mr Trump running again is the "best thing for the party, frankly,” according to Mr Graham.
Meanwhile, on Monday, Mr Trump said that his campaign will join an improbable case before the US Supreme Court challenging election results in Pennsylvania and other states.
His latest move came after the Supreme Court rejected a last-gasp bid to reverse Pennsylvania's certification of US president-elect Joe Biden's victory in the state.
The court has asked for responses by Thursday.
Out of the roughly 50 lawsuits filed around the country contesting the 3 November vote, Mr Trump has lost more than 35 and the others are pending, according to an Associated Press tally.
The suit from the Texas attorney general, Republican Ken Paxton, demands that the 62 total electoral college votes in Georgia, Michigan, Pennsylvania and Wisconsin be invalidated. That is enough if set aside, to swing the election to Mr Trump.
Mr Paxton's suit repeats a litany of false, disproven and unsupported allegations about postal votes and voting in the four battleground states.
Mr Trump said: "We will be intervening in the Texas (plus many other states) case. This is the big one. Our Country needs a victory."
Legal experts dismissed Mr Paxton's filing as the latest and perhaps longest legal shot since election day, and officials in the four states have sharply criticised Mr Paxton.
The Supreme Court refused to call into question the certification process in Pennsylvania. Governor Tom Wolf has already certified Mr Biden's victory and the state's 20 electors are to meet on December 14 to cast their votes for the former vice president.
Mr Biden won 306 electoral votes, so even if Pennsylvania's results had been in doubt, he still would have more than the 270 electoral votes needed to become president.
Additional reporting by Associated Press
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