Trump was initially scheduled to make the announcement in New York, but postponed it in the wake of the terror attack in French city of Nice that killed 84 people and injured over 100.
Officially, Trump has not made any announcement.
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Pence, 57, a lawyer by training and Governor of Indiana since January 2013, has accepted the offer and a formal announcement can be made as early as Saturday, CNN reported, so did other media outlets.
According to reports, Trump told a fund raiser in Los Angeles last evening that he has made his pick, and was ready to announce. Though, he did not identify his pick.
In an interview to Fox News Trump said he has not made a "final decision" yet.
Two other candidates in the race are Newt Gingrich, the former House Speaker and Governor Chris Christie on New Jersey.
"We have to call him. But let me just say this. I have great respect for Governor Pence. But I also have great respect for Newt. I have great respect for Chris Christie. We have some tremendous people in the Republican Party but I will be making the announcement at the appropriate time," he was quoted as saying by Fox News.
Meanwhile, Pence has reportedly abandoned his re-election bid as Indiana Governor.
He was elected as the Indiana Governor in 2013. Before this he was a member of the US House of Representatives for 19 years from 2003 to 2013.
Pence keeps a low-profile and can play a crucial role in uniting the Republican party to rally around the controversial billionaire ahead of the November presidential elections.
He has often described himself as "a Christian, a conservative and a Republican, in that order."
As governor of Indiana, Pence has invited criticism for signing bills making it harder for women to have abortions.
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