"I see that you are on the job and presidential, as always," Biden wrote.
The back-and-forth exchange came hours after the president's son, Donald Trump Jr., retweeted a 15-second mock clip of Biden's response that featured an image of the ex-vice president lurking in the background, eventually placing his hands on his own shoulders and face in the back of his head. The image used for the faux Biden is from the viral photo of him placing his hands on the shoulders of Stephanie Carter, the wife of former Secretary of Defense Ash Carter.
Trump captioned the tweet, "WELCOME BACK JOE!"
The photographs of Biden and Carter made their way back into the public eye following recent claims of Biden's inappropriate behavior toward women. Carter wrote a post on Medium defending Biden and saying the photos were “misleading.”
Biden posted a video on Wednesday to social media in which he acknowledged his tendency toward physical displays of affection and encouragement have made some women uncomfortable. In it, he also promised to be “much more mindful” of respecting personal space.
The video was Biden’s first direct comment on what has tripped up his preparations to enter the 2020 Democratic presidential campaign. He did not directly apologize but seemed to be seeking to ease some people's discomfort, which has raised questions about whether he could wage an effective campaign.
Trump talked about Biden at a National Republican Congressional Committee dinner Tuesday, saying only one who's not a socialist is "being taken care of pretty well by the socialists."
Trump added: "I was going to call him and say, 'Welcome to the world, Joe. You having a good time, Joe?'"
Last week, former Nevada Lt. Gov. candidate Lucy Flores accused Biden of inappropriate conduct during a 2014 campaign event, saying Biden “plant[ed] a big slow kiss on the back of my head.” Multiple other women have also come forward to accuse Biden of similar actions.
Trump has been accused of sexual misconduct by multiple women but has denied it.
The Associated Press contributed to this report.