Republican senators ABC News spoke with Thursday squirmed when asked about former President Donald Trump falsely questioning Vice President Kamala Harris' racial identity during his interview at the National Association of Black Journalists convention a day earlier -- as the former president doubled down on the false attack.

In a social media post Thursday morning, former President Donald Trump shared a family portrait of Vice President Kamala Harris and wrote, "Your warmth, friendship, and love of your Indian Heritage are very much appreciated."

His social media post reiterated his false claim that Harris only emphasized her Asian-American heritage -- something he mentioned during his interview at the NABJ convention on Wednesday.

During the interview, he falsely questioned Harris' race. Harris is the daughter of a Jamaican father and an Indian mother.

"I didn't know she was Black until a number of years ago when she happened to turn Black and now, she wants to be known as Black. So, I don't know, is she Indian or is she Black?" Trump said during the NABJ interview.

He went on to say that "she was Indian all the way, and then all of a sudden she made a turn, and she went -- she became a Black person."

Trump allies in the Senate tried to focus less on the former president's comments at NABJ -- when he called Harris' race into question -- and instead pivoted to what they believe are his policy accomplishments in his first term.

Sen. Lindsey Graham, R-S.C., a close ally of the former president's, said Trump would be better served focusing on policy.

Asked if it was appropriate for Trump to question Harris' race, Graham replied: "No, I don't think so."

"I've known the vice president for a while, she has always embraced her heritage proudly as she should," Graham said. "My problem with Vice President Harris is the policy choices she's made. I think she's live a consequential life, but on policy the country is on fire -- the world is on fire and the country is in decline. And I think we need new leadership so so that will be my approach."

When pushed on whether Trump's comments were productive in pushing voters toward focusing on policy, Graham dodged, saying "I think the way back into power is to compare [Trump's] presidency with what's going on today, offer solutions to problems -- that's the way back into power."

Sen. Thom Tillis, R-N.C., pushed repeatedly on whether Trump's comments were appropriate, conceded that they were likely a distraction.

"I think it shifts away from the discussion I want to focus on, but it may very well be that we have a difference of opinions about what is going to move the voters," Tillis said. "I for one think its the failure on the economy the failure on the border and the failure on national security."

Sen. John Cornyn, R-Texas, who is running to lead the Senate Republican Conference when McConnell steps down this fall, sidestepped questions repeatedly.

"I really don't have anything to say about that. I did think it was relevant that he showed up and she did not and I appreciate the fact he is willing to even appear in front of hostile environments like that one," Cornyn said.

Asked about Trump's apparent confusion about Harris being biracial, Cornyn suggested that "I think we are all a combination of something right?"

House Republican leaders have privately told their conference to focus their attacks against Harris on her record, sources familiar with the conversation told ABC News.

The guidance came after a number of House Republicans made references to Harris' race and gender when asked by reporters about her bid for the White House with some -- such as Rep. Tim Burchett and Rep. Marjorie Taylor Greene -- calling her a "DEI Vice President" or "DEI hire."

Separately, sources also told ABC News, Speaker Mike Johnson also privately told members to focus on drawing a contrast with Harris' record against the "strength" of Trump. Johnson has privately and publicly insisted this has nothing to do with race.

Sources told ABC News that one member said, "pointing out she's not a white man, is not a winning campaign message."

Several Democratic senators condemned Trump's comments about Harris' race.

Senator Elizabeth Warren, D-Mass., said Trump's comments were "disgusting."

She said she admires the way Harris responded to the comments, which Harris on Wednesday night called "the same old show" with "divisiveness and disrespect."

"I really admire what VP Harris said when she said she is ready to turn the page on that and start again with [a] new leader who is completely focused on how we make this government work better for hard-working families," Warren said.

Sen. Raphael Warnock, D-Ga., called Trump's comments a "distraction."

"This is a distraction. And that's what the former president majors in – the politics of distraction and division. But I don't think it will stand. Because Kamala Harris is focused on the people that she wants to represent in this country," Warnock said.

ABC News' Lalee Ibssa, Soorin Kim, Kelsey Walsh, Rachel Scott and Jay O'Brien contributed to this report.