Lance Armstrong must pay a prize-insurer $10 million as punishment for lying under oath about doping, according to an arbitration award decision filed in a Texas state court.

The decision represents a blow for the former pro cyclist who is fighting a couple of legal battles that could strip him of many millions of dollars.

The prize insurer SCA Promotions Inc. asked a Texas state judge to confirm the arbitration award against Armstrong, in a filing Monday. It wants the court to enter a $10 million judgment against Armstrong and former team owner Tailwind Sports, a move which it believes will enable it to collect payment.

“This award is unprecedented,” Armstrong’s lawyer Tim Herman said Monday. The dispute had previously been “fully and finally settled voluntarily” years earlier, Herman said. More recently, Armstrong offered to pay SCA “the entire” $10 million, Herman added, “despite the absence of any legal basis for the sanction.” SCA refused Armstrong’s offer.

SCA’s next step will likely be to question Armstrong under oath in a deposition about his assets, and whether he has enough remaining money to pay the sum, its lawyer Jeffrey Tillotson said.

SCA’s dispute with Armstrong began more than a decade ago, after the former cyclist for the U.S. Postal Service team won the 2004 Tour de France, the sixth of his seven consecutive Tour victories.

Armstrong’s salary was $4.5 million that year, according to his agreement with the team. An addendum to his agreement called for the team to pay Armstrong a $10 million bonus if he won the Tour de France each year from 2001 to 2004. The agreement also required the team to ensure those bonuses.

The team owner purchased coverage from SCA Promotions, a small Dallas, Texas, firm that protects the financial interests of team owners and sponsors. SCA offers “hole-in-one” insurance and other coverage of athlete performance incentives in football, basketball, baseball, soccer, Nascar, and golf.

But SCA refused to give Armstrong $5 million in bonuses, on the suspicion that he had cheated to win, prompting Armstrong and Tailwind Sports to sue SCA for payment. The case went to arbitration, and Armstrong said under oath that he didn’t dope. “I race the bike straight up fair and square,” he testified in January 2006, according to a transcript of his testimony before a panel of three arbitrators, who are experienced lawyers.

Concerned it might lose, SCA ultimately agreed to a $7.5 million payment to settle the original arbitration case. That sum represented Armstrong’s bonuses, interest, and attorney fees. The agreement said

“No party may challenge, appeal or attempt to set aside” the settlement. It also said the agreement was “fully and forever binding.”

But in 2012, Armstrong was stripped of nearly all of his titles, including his Tour wins, following a ferocious battle with the U.S. Anti-Doping Agency. In 2013, he publicly acknowledged that he had used performance-enhancing drugs when he captured his seven Tour victories.

SCA then accused Armstrong of fraud, demanding that he repay it more than $12 million, or the sum it paid him for Armstrong’s Tour wins from 2002 to 2004. Armstrong refused, and his lawyers said that Texas state law doesn’t allow the company to reopen the original settlement.

The arbitration panel agreed to reconsider the case. Two of three arbitrators, representing a majority, issued a “final” arbitration award ordering Armstrong and Tailwind Sports to pay SCA $10 million. “Deception demands real, meaningful sanctions,” the two-person majority wrote in their Feb. 4 decision.

A dissenter among the three arbitrators argued that the $10 million sanction represents “an unwarranted, unlawful reversal” of the original settlement.

Former team owner Tailwind Sports previously asserted that the arbitrators have no jurisdiction over Tailwind because it was dissolved consistent with Delaware law, on Dec. 31, 2007. In any event, the panel would still have jurisdiction over Armstrong.A dissenter among the three arbitrators argued that the $10 million sanction represents “an unwarranted, unlawful reversal” of the original settlement.

Armstrong also faces a pending False Claims Act lawsuit, in which his former teammate Floyd Landis alleged that Armstrong defrauded the U.S. Postal Service as a former team sponsor. Landis filed the lawsuit in June 2010 in order to help the government recover at least some of the $40 million the Postal Service paid to sponsor Armstrong and his cycling team from 1998 to 2004.

“We will continue vigorously to defend Lance against this wrongheaded lawsuit,” said Elliot R. Peters, an attorney with Keker & Van Nest, who represents Armstrong in the False Claims Act case.

—Ashby Jones contributed to this article.