A political spokesman for Kasich declined to comment.
According to the sources, the same Kasich allies who have met with some of the most influential donors in the country have suggested to the governor that there are two scenarios in which he should challenge Trump in a primary.
First, would come after a potential 2018 congressional midterm wave that gives Democrats majorities in the House and the Senate. With that, Republican voters could potentially move toward a candidate like Kasich, who is considered more of a centrist in the GOP. Such a loss in the midterms could also signal to GOP donors that there's a need for drastic change at the top.
Trump's approval rating stands at just lower than 42 percent, according to a polling average calculated by nonpartisan website Real Clear Politics.
The other scenario pitched to Kasich would ride on the political implications of the ongoing investigation by special counsel Robert Mueller. The probe is looking into whether the Trump campaign colluded with Russian operatives during the 2016 presidential election and whether the president obstructed justice in the investigation.
If the investigation makes its way into the Oval Office, Kasich's friends have said, it may be an opportunity for the governor to run as either a Republican or an independent.
This latest development comes as buzz continues to build around another potential Kasich run for the White House. In March, he said "all of my options are on the table" for 2020, according to Politico.
The Ohio governor is also hitting states that are critical to winning presidential primaries. During his visit to New Hampshire earlier this month, he said in an interview with The New York Times that he considers himself a "hybrid" Republican and more people are approaching him since his loss in 2016.
"I have people of all shapes, sizes, philosophies and party preferences that approach me. But what does that mean? I don't know. I'm on television, so all the sudden they want to talk to me. Television moves everybody up, right?" he told the Times.
Charlie Black, a former advisor to Kasich's 2016 presidential campaign, told CNBC that he thinks the scenarios are part of an ongoing discussion and warned that his old boss would not stand a chance against Trump in a primary within the current political climate.
"Trump presently has about an 85 percent job approval among primary voters. Unless that dropped dramatically, no one can compete with him for the nomination," Black said. "He would have to be under 50 before I would advise anyone to run."
