Alienating the Senate majority leader could make it far more hard for Trump to achieve his legislative goals, which include tax cuts and a boost in infrastructure spending as well as health care reform.
Trump assailed McConnell on Twitter both Wednesday and Thursday.
Price said in an interview with Fox News that the "onus is on Congress" to find a solution to rising premiums in the ACA's exchanges, particularly in the wake of Anthem's further exchange exits this week.
Changes to the health care system are looking less and less likely this year.
President Donald Trump on Thursday stepped up criticism of Senate Majority Leader Mitch McConnell for not muscling through a health-care bill, escalating an extraordinary fight with a key leader of his own party.
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The White House also confirmed earlier press reports that the President and McConnell had spoken by telephone on Wednesday before Mr. Trump first tweeted his displeasure with the Senate Majority Leader. Then, he implied that McConnell should step aside from his position as Senate Majority Leader. "After 7 years of hearing Repeal & Replace, why not done?"
Speaking to reporters at his golf club in Bedminster, New Jersey, where he is spending a working vacation, Trump was asked if McConnell should stand down.
"Our new president, of course, has not been in this line of work before", McConnell said, according to ABC affiliate WCPO.
But Trump has not relented on the health care issue and continues to attack McConnell, a popular leader among Senate Republicans.
"More excuses", wrote Scavino, an outspoken Trump loyalist.
Mr. Cuccinelli, meanwhile, said Mr. McConnell - not Mr. Trump - is to blame for raising expectations by vowing to rip Obamacare out "root and branch" and by promising to pass border wall funding by the end of April and tax reform by August. He signed off with a somewhat encouraging message: "You can do it!"