Donald Trump continued to stoke speculation on Sunday that he is poised to shake up his senior staff – at the expense of his chief of staff, John Kelly.

The president said in an interview with Fox News Sunday that he was thinking about changing “three or four or five positions”, but without giving details or a timetable.

The homeland security secretary, Kirstjen Nielsen, is also believed to be especially vulnerable as Trump continues to crack down on migration into the US across the Mexican border.

'A dangerous precedent': Texans outraged at prospect of tent cities for migrants Read more

Trump made clear that he wished she would be tougher in implementing his hardline immigration policies and enforcing border security.

Meanwhile, he told the Fox News Sunday host Chris Wallace in an interview at the White House, pre-recorded on Friday, about Kelly that “there are certain things that I don’t like that he does”. He praised his work ethic. But added: “There are a couple of things where it’s just not his strength. It’s not his fault. It’s not his strength,” adding that Kelly himself might want to depart.

Asked whether he would keep Kelly in his post through 2020, the president offered only that “it could happen”. Trump had earlier pledged publicly that Kelly would remain through his first term in office, though many in the West Wing were skeptical.

Meanwhile, on other Sunday talkshows, critics were gathering. The president must face a challenger from inside his own party in the 2020 presidential election in the name of decency, a retiring Republican senator insisted, as the messy fallout from the midterm elections continued.

Jeff Flake, the outgoing Arizona senator and longstanding critic of Trump, said on Sunday morning: “Somebody needs to run on the Republican side, if nothing else to remind Republicans what it means to be a conservative, what being a conservative really means, what it means to be decent as well.”

Speaking on CNN’s State of the Union, Flake said: “I think that the future of the party is with people with an optimistic vision moving ahead. I don’t think that will be me, but somebody needs to run.”

Sign up for the new US morning briefing

Republicans held on to control of the Senate in this month’s midterms, but lost control of the House.

Flake said he hoped that Trump would face a challenger – name-checking prominent conservative critics such as John Kasich, the Ohio governor, who ran for the nomination last time, and the Nebraska senator Ben Sasse, who has called Trump a “megalomaniac strongman”.

Kasich was in New Hampshire last week, seemingly laying the ground for a run against Trump for the party’s nomination, despite playing it down.

Meanwhile, questions about the state of democracy continue to roil the US after this month’s elections. In Georgia, where Abrams first admitted defeat on Friday evening but vowed to pursue legal action to combat voter suppression, she is preparing a lawsuit against the state of Georgia for gross mismanagement of the election. Much about the race in that state was seen as a reflection of divisions in America as a whole.

Her Republican rival, the former Georgia secretary of state Brian Kemp, was responsible for removing voters from the electoral rolls and systematically making it difficult for people to vote, she said.

“Trust in our democracy relies on believing that there are good actors that are making this happen and he was a horrible actor who benefitted from his perfidy,” Abrams told CNN.

Abrams said democracy had failed in Georgia because of underfunding, disinvestment in election polling and training, dramatic discrepancies in absentee ballots and a “purge of voters”.