Trump's endorsement would come at a rally in Green Bay, Wisconsin, Ryan's home state, the network said.
Trump also threw his support behind Senator John McCain of Arizona and Senator Kelly Ayotte of New Hampshire, two lawmakers facing tough re-election fights, as he pressed for party members to overcome differences.
An aide to Ryan said of Trump's plans: "We have no knowledge of this and it's a question for their campaign".
But in neighbouring Wisconsin, a state Trump insists he can win, the state's best-known Republicans said they were too busy to attend an evening event. He said in the same interview that McCain had not done enough for veterans and criticized Ayotte for distancing herself from him during the campaign. Trump's refusal to endorse Ryan ― using Ryan's own phrase ― was widely viewed as payback for the House speaker's hesitation earlier this year.
Ryan, McCain, and Ayotte all publicly condemned Trump's repeated disparagement of Khizr and Ghazala Khan, declaring their support for Gold Star Families.
Trump's endorsement emerged as he took other steps to get his campaign back on track after days of controversy and falling poll numbers that have given Democrat Hillary Clinton the advantage in the race to the November 8 election.
In the latest Reuters/Ipsos poll released on Friday, Mrs Clinton's lead over Mr Trump narrowed to less than 3 percentage points, down from almost 8 points on Monday.
The New York businessman announced he was setting up an economic advisory team to help guide him on economic policy. The group relies heavily on hedge fund managers and investment bankers, a group Trump has railed against.
PGA Championship: Mickelson frustrated
So Sunday will be a day full of golf as the tournament will try and get close to 36 holes in those at the top of the leaderboard. With that said, being able to come back and birdie 8 and 9 really made things a lot better for me on the back side.
Clinton is expected to deliver her own economic plan to the Detroit Economic Club on Thursday.
Trump conceded Friday he was wrong to claim repeatedly in recent days that he saw a video of a US cash payment going to Iran.
At a rally in Des Moines, Mr Trump mostly stuck to his central charge that Mrs Clinton was the "queen of corruption", and defended himself against her charge that he was temperamentally unfit for the White House.
"I'm not going to try and psychoanalyze this stuff", the House speaker told Weber. Trump's running mate Mike Pence has already endorsed Ryan.
The club says it will instead focus its efforts on "reclaiming the Republican Party from those who have done it considerable harm". He also called Ayotte "weak" and claimed to be beating her in polls, although the two are not competing and the senator has higher favorable numbers than Trump. If I endorse him, I'll get credit for being a team player at a moment when the party is struggling to unify and I'll get to say, implausibly or not, that it was my endorsement that contributed to Ryan's very big win after it happens. "I believe we need Paul Ryan in leadership in the Congress of the United States".
When asked about her claim last week that the Federal Bureau of Investigation had found her statements about her email "truthful", when, in fact, it had not, she said on Friday she may have "short-circuited" her reply.
"He doesn't respect me as a human, not just because I'm Native American, but because I'm a woman", she said.
She maintained that "I never sent or received anything marked classified", while acknowledging that some material she sent may retroactively have been considered classified by other government agencies.