Paul Manafort has resigned as chairman of Donald Trump's presidential campaign, under fire in connection with a Ukrainian corruption investigation and two days after being sidelined in a team reshuffle.

"This morning, Paul Manafort offered, and I accepted, his resignation from the campaign," the Republican nominee said in a statement, paying tribute to the 67-year-old veteran strategist and lobbyist's skills.

Mr Trump had recruited Mr Manafort, who formerly advised the pro-Kremlin former Ukrainian president Viktor Yanukovych, earlier this year to head off what was then a potentially contested Republican convention.

"I am very appreciative for his great work in helping to get us where we are today, and in particular his work guiding us through the delegate and convention process," Mr Trump said.

"Paul is a true professional and I wish him the greatest success."

Trailing Mrs Clinton in national opinion polls, Mr Trump has tried to reset his campaign this week, announcing on Wednesday a shake-up of his senior campaign staff for the second time in less than two months.

In the past week, he has abandoned his free-wheeling style of campaigning, instead using a teleprompter at every rally.

Mr Trump also began adding non-rally events to his campaign schedule, visiting a police lodge yesterday afternoon and hosting a round table interview on Wednesday morning.

Previously, he had eschewed such events that historically comprise a significant portion of a candidate's time.

Earlier, Mr Trump apologised for past remarks that "may have caused personal pain" as he sought to refocus his message in the face of falling opinion poll numbers.

Mr Trump was speaking in Charlotte, North Carolina, in his first speech since shaking up his campaign team.

He said: "Sometimes, in the heat of debate and speaking on a multitude of issues, you don't choose the right words or you say the wrong thing.

"I have done that, and I regret it, particularly where it may have caused personal pain. Too much is at stake for us to be consumed with these issues."

Mr Trump did not cite any examples of such remarks. 

The New York businessman has made his "tough talk" and brash style a selling point of his campaign for the US presidential election, rarely apologising in the face of criticism even from within his own party for comments insulting women, Muslims and Mexican immigrants.

In his presidential announcement speech last year, he described some Mexican immigrants as "criminals and rapists".

He recently faced a barrage of criticism for belittling the family of a Muslim American soldier who died in Iraq in 2004, after the soldier's father spoke out against Mr Trump at the Democratic National Convention last month.

The campaign for his Democratic rival, Hillary Clinton, quickly dismissed his apology, saying: "Donald Trump literally started his campaign by insulting people.

"We learned tonight that his speechwriter and teleprompter knows he has much for which he should apologise," the campaign said in a statement.

"But that apology tonight is simply a well-written phrase until he tells us which of his many offensive, bullying and divisive comments he regrets - and changes his tune altogether."