NBC and CNN television said Edwards is due to make an official announcement on winding up his campaign on Wednesday.
If confirmed, the decision would have come after Kerry won a Democratic primary in the crucial electoral battleground state of Ohio.
According to US television networks, the win was Kerry's first in 10 Super Tuesday contests.
The veteran lawmaker and Vietnam veteran went into Tuesday's showdown hoping to land a knockout blow on his only serious challenger for the Democratic nomination.
And if Kerry emerges as the Democratic nominee as expected, Ohio will likely play a key role in November's election battle against Republican President George Bush.
Meanwhile, US Vice-President Dick Cheney has insisted he will be on Bush's re-election ticket, and has brushed aside speculation the president might consider replacing him.
"He's asked me to serve with him on the ticket again for the next four years," Cheney told Fox News in one of a series of cable television interviews.
Edwards may run as John Kerry's
"I'm happy to do that as long as I can be of assistance and he wants me in that spot, I plan to serve."
Some political analysts have speculated Bush might drop Cheney, who has become a lightning rod for Democratic criticism, if he is down in the polls as the November election approaches.
In recent months, Cheney's image has been battered by attacks on his role in the administration's Iraq policy and previous ties with energy company Halliburton, which is being probed for alleged overcharging for its services in Iraq.
Cheney told MSNBC if he thought he had become a liability to Bush, he would tell him to find another candidate.
"Obviously, if a health question came up or something like that, clearly if I thought that I couldn't do what he needs to have done, then I'd be the first to recommend that he gets somebody else," Cheney said.
But he said that was not the case. Cheney has a history of heart trouble including four heart attacks, but says he is in good health.
The vice-president's popularity has fallen and more than one in four Republican primary voters want Bush to pick a new running mate, according to a new poll by the Annenberg Public Policy Center of the University of Pennsylvania.
Thirty-six per cent of Americans held an unfavorable view of Cheney in February, up from 26% in October.
Among likely Republican primary voters, 62% said Bush should keep Cheney, while 27% thought he should be dumped.
Cheney was a prime mover in the decision to invade Iraq and spoke repeatedly of a danger posed by Iraqi weapons of mass destruction. No such weapons have been found.
Cheney's reluctance to answer media questions re-emerged on the topic of Bush's decision to back a constitutional amendment to ban gay marriage.
Cheney's daughter Mary, a senior official in the Bush/Cheney campaign, is openly gay.
The vice-president had said in the 2000 campaign that he did not necessarily see a role for the federal government in the gay marriage issue.
"He (Bush) has asked me to serve with him on the ticket again for the next four years. I'm happy to do that as long as I can be of assistance and he wants me in that spot, I plan to serve"
Cheney was asked by MSNBC if he is ever conflicted between his role as a father and that of a politician.
He said he considers his daughters' lives private "and I think that's the way it ought to remain".
And he told CNN, "I support the president," but said any advice he gave Bush on the gay marriage issue was private.