The biggest creditor of iHeartMedia is close to blinking in its nearly six-month stare-down with America’s largest owner of radio stations, The Post has learned.

Mutual fund Franklin Resources is working with investment bank PJT Partners on a restructuring plan that could save iHeart from bankruptcy — although it may ask for more cash guarantees and possibly a controlling equity stake in return, a source close to the situation said.

Franklin is preparing a proposal to present to fellow creditors and iHeart — formerly known as Clear Channel — as early as next week, a source close to the situation said.

Franklin, which owns $2.3 billion of iHeart’s $20 billion in debt, has until now taken the hard stance that it could get a better recovery in bankruptcy than by accepting a restructuring.

But Franklin’s adviser, PJT Partners, has recommended not pushing iHeart into a bankruptcy, a last-ditch scenario which likely would result in a sale of the most profitable of its 855 stations, which include New York City’s Z100 and 103.5 KTU, the source said.

In addition to keeping iHeart’s best stations, Franklin’s restructuring plan also would likely leave in place iHeart Chairman and CEO Bob Pittman, the former MTV honcho who is generally liked by creditors, the source said.

In exchange, however, sources said Franklin may ask for creditors to get a majority stake in the business. Also, it might ask iHeart to issue new debt, perhaps as much as $1 billion, that would need to be paid first before equity holders could take any distributions or dividends, the source said.

There remains a possibility if Franklin’s ask is too stiff, iHeart will simply reject the offer and go bankrupt.

Still, if it makes a truce with iHeart private equity owners Bain Capital and THL Partners, Franklin would be taking a softer approach than David Tepper’s Appaloosa Management, which successfully stuck to its demands when battling Apollo Global Management’s Caesars Entertainment last year.

On Thursday, iHeart said it priced a $150 million bond offering that pays 8.75 percent interest so it can stay current with debt payments over the next several months.

“I don’t think iHeart would be raising these bonds unless they thought they were going to reach a restructuring deal,” the source said.

Last month, several lenders to iHeart split from Franklin and came close to reaching a new restructuring plan with iHeart. However, the creditors, led by Symphony Asset Management, will not sign the deal unless Franklin ultimately agrees to terms, sources said.

Franklin is not yet in discussions with the Symphony group and likely will offer an alternative restructuring, the source said.

The Symphony-led group’s proposal calls for them to gain control of 49 percent of iHeart — with exiting equity holders retaining 51 percent of the value, a regulatory filing revealed.

The Symphony-led creditors have proposed there be $500 million in new debt that would need to be repaid first, before equity gets any money.

Shares of iHeart on Thursday were up 21 percent, to $1.75, on the news of the bond pricing.

Franklin declined comment, and PJT did not return calls.