McConnell had accused Trump of having "excessive expectations" for legislative progress during remarks to a Rotary Club in Kentucky on Monday.
Shortly before a week-long July 4 recess, Trump tweeted that Republicans should simply repeal the Affordable Care Act rather than replace it - just as McConnell was seeking to reel in wayward conservatives to a broad rewrite of the health law.
The escalating attacks, now in a second day, underscore the rocky relationship between the White House and what typically would be the president's allies from his party in Congress.
However, when reporters asked whether McConnell should resign, Trump only told them to ask should the majority leader fail to get several legislative items passed.
After his last defeat on healthcare, Mr McConnell said the Senate would move on to other matters. The House of Representatives approved its version in May, but its Senate failure - thanks to defecting Republican Senators - marked the collapse of the party's attempt to deliver on vows to erase Obama's law. The report said Trump became "animated" and implored McConnell to continue to push for a repeal of Obamacare. Both before and after taking office, Trump spoke often about passing the massive health care overhaul quickly.
He added that 52 is "a challenging number", a reference to the GOP's scant 52-48 Senate majority.
Republicans placed health care at the top of their 2017 agenda when the year began.
"You can see the president's tweets", said White House Press Secretary Sarah Huckabee Sanders about the Twitter messages for Sen.
The early morning tweet is a continuation of the president's singling out of the one man tasked with leading the GOP in the Senate.
"The votes are simply not there", McConnell said earlier this month.
The blame game between the president and McConnell came as opinion polls this week showed public approval rating for Congress as well as for Trump dipping to new lows.