In an interview with CNN's Ana Cabrera, Dole -- the 1996 Republican nominee who said he spoke with Trump last week -- also suggested Trump apologize to Arizona Sen. John McCain and New Mexico Gov. Susanna Martinez for disparaging comments he's made about them in the past.
"I have been a Republican all my life," said Dole, who endorsed former Florida Gov. Jeb Bush at one point during the 2016 primary fight but has since backed Trump. "The party has done a lot for me. I hope I've done a lot for the party along the way. But when Donald Trump clinched the Republican nomination, it was an easy call for me. What's a lifelong Republican supposed to do, support the opponent? I don't think so."
Asked about notable Republicans who haven't yet supported Trump -- including Bush and House Speaker Paul Ryan -- Dole said, "Well, they still have time to get on the train. It's not moving that fast right now."
Dole pointed out that at one time, Bush had said he would support the GOP nominee, regardless of who it was.
"You know I believe Jeb is a man of integrity and honesty. I just hope he keeps his word when he said he would support the nominee," Dole said. "I know Trump didn't make it very easy for him, because of all the things he said about Jeb, but Jeb is bigger than that and I do hope to see him on board. It would mean a lot in states like Florida and Jeb has friends all over the country. So does his dad and brother and mother."
Gingrich for VP
As Trump searches for a vice presidential candidate, Dole suggested the political novice will need someone well-versed in the ways of Washington.
"My view is that Donald Trump needs someone who understands Congress, who can help him work with Congress, who understands foreign policy, domestic policy, economic policy. You know someone like Newt Gingrich," Dole said. "You know none of us are perfect, but Newt Gingrich is a good fit for Trump, because he can help him in all of those areas and Trump has to listen."
Dole added that Gingrich, who was a controversial House speaker during the 1990s, is someone Trump could trust and a man who would have the courage to tell the mogul when he's made mistakes.
"He would be a big, big help to Donald Trump, because he's going to need it. He has never been a legislator. He has never been elected to anything," Dole said.
Trump rarely -- if ever -- issues public apologies, but Dole said some mea culpas from the presumptive GOP nominee were in order. First, Dole said Trump should apologize to McCain for his remarks last summer that the Arizona senator isn't a war hero because he was captured during the Vietnam War.
"One of my best friends, closest two or three friends, is John McCain," said Dole, who was permanently wounded in action during World War II. "I am going to try to get Trump to issue an apology. I mean, John McCain suffered -- tortured, broken arm, what do you expect? This young man at the time did everything that you would want a serviceman to do, above and beyond the call of duty."
On Sunday the Trump campaign declined comment about Dole's suggestion that the presumptive Republican nominee issue an apology to McCain.
Dole also commented on Trump's insulting Martinez by blaming
her for a growing number of food stamp recipients and for New Mexico's response to the federal government's resettlement of Syrian refugees in the state.
"There is nothing wrong with apologizing to the governor of New Mexico or John McCain of anybody who he has insulted," Dole insisted.
He added that although apologies to Mexicans, Muslims and women for controversial comments he's made in the past "are not necessary, but they are the right thing to do."
'Bring the party together'
Dole said he believes Trump still has time to win over women, minorities and other groups with whom the presumptive GOP nominee is polling badly -- if he's willing to tone down his rhetoric.
"I think there is still time. It is not too late. I can already see sort of a shift with Trump," he said. "He needs to start talking (like) he is about to be president."
Dole said he doesn't "applaud (Trump) for the insults."
"I, as a supporter, I would like him to issue a blanket apology or an individual apology. These people don't deserve that. They are good, strong, Republicans," Dole said.
He continued, "That's how we bring the party together. We want to bring the party together to try and reach out to those that you have offended."