Trump, according to the book "Fire and Fury" by Michael Wolff, "probably prefers the notion of more people having health insurance than fewer people having it."
"He was even, when push came to shove, rather more for Obamacare than for repealing Obamacare," Wolff writes in describing Trump's mindset after winning election in November 2016.
"In fact, he probably favored government-funded health care more than any other Republican," said Wolff in his book, which has been roundly denounced by the Trump administration.
Republicans have adamantly opposed Obamacare since before it became law in 2010.
And no government-funded health-care program other than Medicaid, which covers 68 million people, provides coverage to more people than Medicare.
The Medicare program currently covers nearly 59 million beneficiaries, most of them age 65 and older.
Sen. Bernie Sanders, I-Vt., and other progressives have called for Medicare to be expanded to cover every American, replacing the current health-coverage system, which is a hodgepodge of private and public insurance plans.
Obamacare is just one, relatively small part of that system.
The health-care law since 2013 has expanded coverage to about 20 million people, primarily by expanding access to Medicaid, which covers primarily poor adults, and by subsidizing the purchase of private insurance plans for others.
While Obamacare was promoted by a Democratic president, Barack Obama, and approved by a Democratic-controlled Congress, many on the left were disappointed the law did not achieve their goal of universal health coverage.
More than 12 percent of U.S. adults still lack health insurance, despite Obamacare.
According to Wolff's book, Trump after winning election appeared open to the idea of using an existing government program other than Obamacare to reduce that uninsured rate.
"'Why can't Medicare simply cover everybody?' he had impatiently wondered aloud during one discussion with aides, all of whom were careful not to react to this heresy," Wolff wrote.
The book says that Steve Bannon, a top Trump advisor at the time, "held the line, insisting, sternly, that Obamacare was a litmus Republican issue, and that, holding a majority in Congress, they could not face Republican voters without having made good on the now Republican catechism of repeal."
Bannon believed repeal would "be the most satisfying" policy goal for the new Trump administration.
"It would also be the easiest one to achieve, since virtually every Republican was already publicly committed to voting for repeal," the book says.
Wolff writes that Trump was visited at his estate in Bedminster, New Jersey, by House Speaker Paul Ryan, R-Wis., and then Rep. Tom Price, R-Ga., for a briefing on a plan for repealing and replacing Obamacare with a new law.
"The two men summed up for Trump — who kept wandering off topic and trying to turn the conversation to golf — seven years of Republican legislative thinking about Obamacare and Republican alternatives."
Trump ended up endorsing the repeal-and-replace effort, which despite Bannon's confidence repeatedly failed to win approval by Congress last year.
Price became Trump's secretary of Health and Human Services, but resigned in September after exposure of his habit of taking pricey charter jet flights at taxpayers' expense instead of flying commercial.