Early results suggested Mr Trump had won around 81 per cent of the vote, after 13 per cent of ballots had been counted.
Despite Nikki Haley’s decision to drop out of the primary after Super Tuesday on March 6, the South Carolina governor won around 16.5 per cent of the vote.
The results later on Tuesday night are expected to formally establish Mr Trump and Mr Biden as the two main parties’ nominees for November’s election.
Mr Trump currently has 1,089 pledged delegates for the Republican nomination, of the 1,215 required to win. He must win 137 of the 161 delegates available to declare victory on Tuesday.
The primary season has concluded more quickly than usual, due to Mr Trump’s dominance of the Republican race.
Although incumbent presidents usually win the nomination of their party without serious challengers, most opposition parties see a contest between candidates who have not served in the White House.
Mr Trump is the second president in US history in 100 years to be ousted from office and return at a future election, giving him the status of a de-facto incumbent.
Ms Haley was the last of Mr Trump’s rivals to drop out of the race, after Ron DeSantis, Chris Christie, Vivek Ramaswamy and Tim Scott suspended their campaigns.
Most Republican candidates have gone on to endorse Mr Trump, although Ms Haley urged her supporters to “always make up your own mind”.
Welcoming the results on Tuesday, Mr Biden’s campaign released a statement from the president emphasising his message that Mr Trump’s campaign is based on “resentment, revenge, and retribution that threatens the very idea of America”.
“Voters now have a choice to make about the future of this country,” he said.
“Are we going to stand up and defend our democracy or let others tear it down?
“Will we restore the right to choose and protect our freedoms or let extremists take them away?
“Will we finally make the wealthy pay their fair share in taxes – or will we allow corporate greed to run rampant on the backs of the middle class?”
Mr Biden and Mr Trump’s early status as presumptive nominees will increase pressure on the President to debate his predecessor before the scheduled presidential television debates in the autumn.
Mr Trump has already called for an event to be organised and said he would attend even if it was run by the “corrupt” Democratic National Committee.