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Liberty Media already controls Sirius XM. Now, it wants to own the entire company.

On Friday, Liberty proposed to acquire the 48 percent of Sirius it does not already own in an all-stock deal valued at more than $10 billion, making the satellite radio provider a subsidiary of the media conglomerate.

The complex deal would be a capstone to what has already proved to be one of the savviest media deals in recent years, and is just the kind of financial engineering that has come to define Liberty and its chairman, John C. Malone.

In the depths of the financial crisis, Liberty made a multimillion-dollar loan to Sirius in exchange for a large equity stake. In recent years, Liberty has increased that position, and in 2012 took full control of the company, forcing out its former chief executive, Mel Karmazin.

Along the way, the value of Sirius skyrocketed, making Liberty’s initial investment of less than $1 billion worth more than $10 billion today. With the proposal to take full control, Liberty could soon be the sole owner of a company valued at $21.5 billion.

Under the proposed deal, Sirius stock would be converted into new class C shares of Liberty stock. The class C shares would then be distributed to existing Liberty shareholders on a two to one basis. Once completed, Sirius’ public shareholders would own about 39 percent of outstanding common stock of Liberty Media.

“This is not a change of control,” Liberty’s chief executive Greg Maffei said in a call with analysts. “We already control Sirius XM.”

Mr. Maffei said the deal would simplify Liberty’s capital structure and clear up any questions about its commitment to Sirius XM. The deal would also increase the financial flexibility of both companies, freeing them to pursue other deals, he said.

“We think we’re offering them premium,” Mr. Maffei said in an interview. “And we think these C shares will be far more liquid than the Sirius shares.”

The Sirius board will have to form a special committee to evaluate the deal. That special committee could ask for a higher price.

But the transaction would be tax free for both Sirius and Liberty, as well as the shareholders of both companies, making it yet another tax-efficient deal that Mr. Maffei and Mr. Malone have been able to engineer.

This is not the only big transaction Liberty is working on. Charter Communications, a cable operator in which Liberty has a large stake, is plotting a multibillion-dollar bid for Time Warner Cable. In a call with analysts, Mr. Malone said any deal between Charter and Time Warner Cable would probably close much later than a potential deal for Liberty to buy Sirius.

But taking control of Sirius could, in fact, help Liberty get that deal done. Should Charter bid for Time Warner Cable, Liberty could be called on to provide additional capital for the deal, which could top $40 billion.

“This transaction would give us incremental capital to put into a deal,” Mr. Maffei said in an interview.

With full control of Sirius, Liberty Media would be in relatively uncharted territory: operating a big media company. For the most part, Liberty has opted to own stakes of companies rather than take full control.

But if the Sirius deal got done, it could set up another transaction sometime down the line. In the call with analysts, one questioner asked if Liberty was already thinking about spinning out Sirius as a new public company.

Mr. Maffei responded that it was an unusual time to be asking that question, but that nothing could be ruled out.

Should Liberty go that route, it would fit the pattern established over many years. Among the companies Liberty has spun out are Discovery Communications, Starz and DirecTV.

“The proposed transaction is an important step in the growth of both companies,” Mr. Malone said in a statement.