The new offer of $39.20 per share, or about $19.45 billion, was initially rejected by the board two weeks ago. The board approved the deal once the private equity partners agreed to allow shareholders to own shares in the new company with the same benefits as the equity firms.
The buyer's group is also assuming $8 billion in debt.
A shareholder vote on the new deal has not yet been scheduled, but analysts believe the offer will win over the holdouts.
"This seems like an outcome that satisfies everyone," said Stanford Group analyst Fred Moran. "To us, this looks like a done deal."
Shareholders who want cash can take it, those who want to continue owning shares can do that and the private equity group still enjoys at least 70% of whatever future growth it can get out of the private company, he noted.
Up to 30% of the new company could be owned by current shareholders if the deal gets approval from two-thirds of them. A vote will be scheduled after Clear Channel meets regulatory filing requirements.
Shares of the San Antonio-based company rose 44 cents, or 1.16%, to $38.23 Friday.
It was not immediately clear Friday whether the Clear Channel holdouts -- including the largest shareholder, Fidelity Management & Research -- would support the new offer, but a source familiar with the position of Highfields Capital Management, the third largest shareholder at about 5%, said the fund would likely support the new offer. The source spoke on the condition of anonymity because the fund was awaiting final details on the proposal before committing to a position.
Because Texas law requires two-thirds of shareholders to approve a buyout offer, activist shareholders enjoyed considerable leverage in the Clear Channel deal, which was first announced in November.
Several had insisted the company was worth more than earlier offers of $36.70 and $39 per share.
The offer to allow shareholders to keep part of the newly private company is an unusual concession because private equity buyers pay a premium to get total control and fewer regulatory requirements than those required of public companies.
Two weeks ago, Kohlberg Kravis Roberts and GS Capital Partners made a similar shareholder concession in their bid for upscale audio equipment maker Harman International Industries.
But Lawrence Hamermesh, a law professor at Widener University's School of Law in Delaware, said it's unlikely such offers will become commonplace.
They are designed to placate shareholders who are willing to strike down deals in favor of waiting out higher profits over the longer term, he said.
"Someone buying out a public company expects to make a lot of money and the public stockholders want a little piece of that action before they'll go along so they don't feel like chumps," Hamermesh said.
Clear Channel is the nation's largest radio-station operator, and it owns 90% of a billboard business that is the world's largest, with 973,000 signs.