The broadcaster made the comments during an interview with Fox & Friends.
Mr Morgan acknowledged that he had been critical of Mr Trump's handling of the coronavirus pandemic in the US but he asked that the president give him a second chance on social media.
“If the president is watching - he might be - I know you un-followed me on Twitter because I was very critical on you in this crisis," said Mr Morgan.
"You may want to re-follow me now because if you take my advice you still have a chance to keep in the White House."
The co-host of Good Morning Britain also addressed a story about Joe Biden's son as he accused American journalists of being "hyper-partisan" for ignoring the story.
A New York Post story claimed Hunter Biden introduced his father to a top executive at a Ukrainian energy firm before the former vice president lobbied for the firing of a prosecutor who had investigated the company.
The story, which focused on an email from 2015 in which a Burisma board adviser thanked Hunter for inviting him to a Washington meeting with his father, was headlined: "Smoking-gun email reveals how Hunter Biden introduced Ukrainian businessman to VP dad."
The Biden campaign said no such meeting ever took place.
Mr Morgan told Fox & Friends: "Journalists should have a duty if they're impartial to investigate it with the same aggression and enthusiasm they would if the name Biden was switched to Trump, and it's not been done."
He added: We saw over the whole Russia collusion fiasco over several years, that they don't need much of this so-called hard evidence to obsess about a story if they want to.
"So they're making a political characterisation, not a journalistic one."
It comes as more than 54 million Americans have cast early ballots at a record-setting pace ahead of the November 3 election, according to data from the US Elections Project.
The rush to vote is a sign of the intense interest in the contest between Mr Trump and Mr Biden, as well as concerns about avoiding crowded polling places on Election Day and reducing the risk of exposure to Covid-19, which has killed more than 224,000 Americans.