NEW YORK (CNN/Money) - News that Halliburton and Honeywell are close to settling multimillion-dollar asbestos-related lawsuits against them sent shares of both stocks higher late Wednesday.

Halliburton Co. (HAL: Research, Estimates), a Dallas-based energy company, said a court extended a temporary restraining order against more than 200,000 pending asbestos claims.

In addition, Bloomberg News, citing plaintiffs' lawyers, reported that Halliburton agreed to pay as much as $4.2 billion to settle asbestos-related claims against several current and former units and wipe out future asbestos liability.

Similarly, Honeywell International Inc. (HON: Research, Estimates) is close to finalizing a deal to resolve "a significant portion" of its asbestos lawsuits, according to Dow Jones, which cited comments from Honeywell attorneys to federal bankruptcy judge Judith Fitzgerald in Pittsburgh.

Honeywell, the Dow Jones report said, has reached agreement with plaintiff attorneys representing about 190,000 of 200,000 claimants in lawsuits related to its former North American Refractories unit, known as Narco.

Halliburton shares rose 83 cents, or 4 percent, to $21.23 in after-hours trading, while Honeywell, an aerospace company based in Morristown, N.J., gained 44 cents, or 1.8 percent, to $25.25.