The Trump camp made that offer originally to John Kasich, Governor of Ohio. They got turned down. The offer was to let Trump's VP pick be responsible for all "domestic and foreign" policy.
Not incidentally, I posted earlier about how Trump might win but has no interest in doing the job.
In May, Paul Manafort, Trump’s campaign chairman, sat down with the Huffington Post’s Howard Fineman for a lengthy interview, which touched on the search for a running mate.
“He needs an experienced person to do the part of the job he doesn’t want to do,” Manafort said in reference to Trump. “He sees himself more as the chairman of the board, than even the CEO, let alone the COO.”
The similarities between the two reports are striking. Trump, who’s never demonstrated any meaningful knowledge of any area of public policy, or really any interest in how government works at any level, apparently wants to be president.
He does not, however, want to do the things presidents are elected to do. That requires hard work, long hours, and, you know, reading and stuff. Trump can’t be bothered with such unglamorous tasks – he wants to “make America great again” while having others roll up their sleeves.
So if Trump wins, the functional president will be radical right-winger Mike Pence.
UPDATE: And how does Trump and his staff respond to these reports? AZBlueMeanie reports denial ("John Kasich was never asked by me to be V.P.") and trash-talk ("Manafort called Kasich an 'embarrassment' "). How else? The Blue Meanie responds to Trump's denial:
Too many sources in a position to know say you did. It’s all just a scam by the biggest con man and grifter in America.