Graham, 65, fought off a strong challenge for his Senate seat on Tuesday, defeating Democrat Jamie Harrison with 54.5 per cent of support versus 44.2 per cent for Harrison.
Graham first entered Congress in 1995, and has known Biden – who has been in Washington DC since 1973 – for many years.
The South Carolina senator said he would work to support Biden, a six-term Delaware senator, where possible.
Graham held a Zoom call with supporters on Friday to discuss his and Trump’s campaign
Lindsey Graham, re-elected on Tuesday to the Senate, said he would work with Biden if elected
Graham and Biden, pictured in January 2015, have known each other for decades
Graham said that Biden ‘deserves’ to have his choice of Cabinet appointees, if elected
‘I ran on two things: that I will be a reliable vote to stop the most radical agenda being pushed by Nancy Pelosi and others in the history of the United States, and I also talked about openly in my campaign at all the debates about how I have reached across the aisle and will continue to do so,’ he said.
Graham, in a Zoom call reported by The Hill, said he’d oppose the Green New Deal, Medicare for All and packing the Supreme Court but would remain open to working with Biden as he appoints his cabinet, if the Democratic candidate is elected.
‘When it comes to finding common ground, I will do that,’ he said.
‘The vice president deserves a cabinet.
‘I will give him my input about who I could vote for as secretary of state, attorney general.
‘There may be some people that I just can’t vote for because I think they’re unqualified or too extreme,’ he said.
Graham also added, ‘I’m not conceding he’s going to win.’
Graham has previously praised Biden’s wife Jill as ‘an outstanding person who has lived a consequential life’.
Biden is pictured speaking to the nation on Thursday night, as vote counting continued
Speculation has been growing as to who Biden would appoint to his Cabinet, and how the fact that the Republicans could control the Senate will affect his thinking.
It now looks likely that the balance of the Senate could be decided in Georgia in January, where run-offs will be held for two seats.
Names being floated include Chris Coons, the Delaware senator, for secretary of state; Pete Buttigieg as UN ambassador, and Michele Flournoy for defense.
Andrew Cuomo, governor of New York, is among the names being mentioned for attorney general, and it would be fascinating to see if the tough-talking New Yorker would be classed by Graham as ‘too extreme’.