Its delegate count has him reaching the quota thanks to a small number of the party’s unbound delegates.
They told AP they would support him at July’s Republican convention.
In North Dakota on Thursday, Mr Trump thanked 15 delegates from the state who he said “got us right over the top”.
He defeated 16 other Republican contenders and reportedly has 1,238 delegates, one more than needed.
Republicans are expected to finalise their nomination at the convention in Cleveland in July.
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US election 2016: Primaries, caucuses and delegates
If confirmed, Mr Trump will face former US Secretary of State Hillary Clinton or Vermont Senator Bernie Sanders, who are vying for the Democrat nomination.
On Wednesday, the New York billionaire suggested going against Mr Sanders in a TV debate in California before the state’s primary on 7 June.
Mr Sanders agreed to the debate in a tweet, saying “Game on”.
Analysis: Anthony Zurcher, North America Reporter
It wasn’t a matter of if, only when. With no real obstacles between him and the nomination, Donald Trump was going to cross over the magic 1,237 delegate mark at the latest by the California and New Jersey primaries on 7 June.
It must be a bit of delicious irony for the New York real estate mogul, however, that the Associated Press has declared him the winner thanks to the support of a Republican Party establishment that largely recoiled from him for most of the campaign.
Of course the nomination isn’t official until the balloons drop at the Republican convention in July, but the desperate attempts of the #NeverTrump movement to throw any obstacles in his path are essentially extinguished.
While his closest presidential rivals – Ted Cruz, John Kasich and Marco Rubio – have yet to free delegates pledged to support them at the convention, Mr Trump can win the prize with or without their help.
The Republican convention in Cleveland will be the Donald Trump show, and everyone not with him will be spectators or – as his recent criticism of Republican Governor Susanna Martinez of New Mexico has shown – targets