WASHINGTON — Kellyanne Conway, a top adviser to President-elect Donald J. Trump, on Sunday assailed Mitt Romney, a leading contender for secretary of state in the Trump cabinet, accusing him of having gone “out of his way to hurt” the president-elect during the Republican primaries.

Ms. Conway’s criticism of Mr. Romney, on ABC’s “This Week” program, came as Mr. Trump is weighing whether to choose Mr. Romney, Rudolph W. Giuliani or perhaps another candidate for the State Department post.

Asked about her comments on Twitter last week that she had received a “deluge” of concern about Mr. Romney, Ms. Conway said she had discussed the issue privately with Mr. Trump and would respect his decision. But she made clear that she personally opposed choosing Mr. Romney as secretary of state.

“There was the Never Trump movement, and then there was Gov. Mitt Romney,” she said on ABC, adding later: “I only wish Governor Romney had been as critical of Hillary Clinton” during the general election. During the primaries, Mr. Romney called Mr. Trump a “fraud” and a “phony.”

Ms. Conway said it was important for Mr. Trump to seek to unify the Republican Party by making gestures to those who opposed his candidacy. But, she added, “I don’t think the cost of party unity has to be the secretary of state position.”

Moments after appearing on the show, Ms. Conway, who is under consideration to be Mr. Trump’s press secretary, wrote in a message on Twitter that she had told Mr. Trump her opinion privately, “and I’ll respect his decision.”

“Point is the volume & intensity of grassroots resistance to Romney is breathtaking,” she wrote.

And on NBC’s “Meet the Press,” Ms. Conway said people felt “betrayed” by the idea that Mr. Romney could get a top job in the cabinet. “I’m not campaigning against anyone,” she said. “I’m just a concerned citizen.”

“We don’t even know if he voted for Donald Trump,” she added.

Her comments came as advisers to Mr. Trump inside the transition effort and outside it are feuding over which direction to go in making one of the most critical appointments to the president-elect’s national security team. Those private differences spilled into public view last week.

A dinner guest who attended Mr. Trump’s Thanksgiving festivities at the Mar-a-Lago resort in Florida said that the president-elect had spent some of the dinner soliciting advice about his choice for secretary of state. A close aide to Mr. Romney said on Sunday that Mr. Romney, the 2012 Republican presidential nominee, had given little indication of his thinking on the matter.

Mr. Trump and his aides also continued on Sunday to harshly criticize a recount effort being undertaken by Jill Stein, who was the Green Party candidate for president. In a series of Twitter posts starting early Sunday, Mr. Trump condemned Mrs. Clinton, whose top lawyer for her presidential bid said over the weekend that the campaign would participate in a recount in Wisconsin and potentially in Michigan and Pennsylvania.

In an initial message at 7:19, Mr. Trump wrote that “Hillary Clinton conceded the election when she called me just prior to the victory speech and after the results were in. Nothing will change.”

He then went on to quote a comment by Mrs. Clinton during one of their debates, in which she said she was horrified by Mr. Trump’s refusal to say that he would accept the outcome of the election. And he noted that in her concession speech, she urged people to respect the vote results.

“‘We have to accept the results and look to the future, Donald Trump is going to be our President,’” Mr. Trump wrote on Twitter, quoting Mrs. Clinton. “‘We owe him an open mind and the chance to lead.’ So much time and money will be spent — same result! Sad.”

Marc Elias, who served as Mrs. Clinton’s campaign lawyer, wrote in a post on the self-publishing platform Medium on Saturday that campaign officials had found no “actionable evidence” of hacking or attempts to tamper with the vote. But he said the campaign would “participate” in the recount effort begun by Ms. Stein.

“We do so fully aware that the number of votes separating Donald Trump and Hillary Clinton in the closest of these states — Michigan — well exceeds the largest margin ever overcome in a recount,” Mr. Elias wrote, adding that it was important to have the campaign represented during any legal proceedings related to the election.

Advisers to Mr. Trump also responded on Sunday to the death of Fidel Castro, insisting that Mr. Trump would take a hard look at President Obama’s diplomatic opening to Cuba.

“He is open to any number of possibilities,” Ms. Conway said on ABC, noting that the president-elect was “open to researching and in fact resetting relations with Cuba.” But she added that Mr. Trump would also look at the possibility of reimposing trade and travel restrictions if that would help get political prisoners in Cuba released.

Reince Priebus, who will be Mr. Trump’s White House chief of staff, said on “Fox News Sunday” that the president-elect would demand concessions from Cuba in exchange for continued improvements in relations.

“Repression, open markets, freedom of religion, political prisoners — these things need to change in order to have open and free relationships,” Mr. Priebus said. “There’s going to have to be some movement from Cuba in order to have a relationship with the United States.”