The bidders, Bain Capital Partners and Thomas H. Lee Partners, previously offered $39 a share.
But shareholders have since been pressuring Clear Channel to reconsider its decision, with several stockowners telling Reuters they had called the company's board to persuade them to delay the vote.
While the new bid appeared to gain more shareholder support, it was unclear whether it would gather enough to succeed.
The deal faces a difficult hurdle because under Texas law two-thirds of the company's shares must approve the transaction. Shareholders who fail to vote are counted as voting against the deal.
The revised structure had been supported by Highfields Capital Management LP, a significant shareholder that originally opposed the bid, sources told Reuters last week.
However it was still unclear whether Fidelity Investments, a major shareholder that has steadfastly opposed the sale, would support the new bid.
Fidelity was not immediately available for comment, although a source close to the mutual fund firm familiar with the situation said on Friday it would still oppose a bid at $39.20.
Opposition from Fidelity and Highfields as well as from proxy advisory service ISS to the original deal had dealt a large blow to the prospects of getting it through a vote.