The sparring kicked off Thursday afternoon, coming shortly after Trump's tweet attacking Obama's endorsement.
"Obama just endorsed Crooked Hillary. He wants four more years of Obama—but nobody else does," Trump tweeted.
"Delete your account," the presumptive Democratic nominee fired back. The message was not signed "-H," which is how Clinton usually signs tweets she authored, suggesting this specific tweet was written by Clinton's staff.
Trump returned fire later Thursday, citing Clinton's controversial use of a private email serve while serving as secretary of state, in his attack.
"How long did it take your staff of 823 people to think that up--and where are your 33,000 emails that you deleted," he tweeted.
Clinton's tweet also had other cheeky Republican detractors. The Republican National Committee's communications director, Sean Spicer, tweeted, "Like with a cloth?" a reference to an off-hand comment Clinton had previously made dismissing criticism of her use of a private email server while serving as secretary of state that was eventually wiped.
And Republican National Committee Chairman Reince Priebus tweeted "if anyone knows how to use a delete key," it's Clinton.
But the morning after Massachusetts Sen. Elizabeth Warren endorsed Clinton, she reinforced the presumptive Democratic nominee's original command to Trump with "No, seriously -- Delete your account."
Obama endorsed Clinton in a web video Thursday.
"I want those of you who have been with me since the beginning of this incredible journey to be the first to know that I'm with her," Obama continued. "I am fired up. And I can't wait to get out there and campaign with Hillary."