Capito, fresh off her re-election to a seventh term in the House, is expected to officially announce her candidacy Monday -- her 59th birthday -- in Charleston, West Virginia MetroNews reported.
Rockefeller, 75, who has not said whether he will go after a sixth term, said in a statement that Capito gave him a heads-up about her intentions, the Washington publication Politico reported.
"Congresswoman Capito called last week to let me know of her plans and I appreciated that," Rockefeller said. "But my total focus right now is on the national budget situation and the fight for West Virginia families -- making sure the very wealthy finally start paying their fair share again, for the first time in decades, rebuilding a strong middle class, and creating real opportunity for those who are still struggling.
"Beyond that big question, everyone I talk to in West Virginia is tired of the non-stop campaigning. West Virginians just want us to do our jobs, and for me that means focusing full-time on the serious issues at hand. Politics can wait."
The Washington publication Roll Call reported a Republican source had confirmed the West Virginia MetroNews report breaking the news that Capito would run.
Roll Call noted Capito, who represents West Virginia's 2nd District, chose not to run against Democratic Sen. Joe Manchin III this year after likewise forgoing a race against Manchin in 2010 in a special election to fill the vacancy caused by the death of Sen. Robert C. Byrd.
MetroNews reported Capito, a moderate, was expected to say the timing is now right for her to run for the Senate.
Capito's father, Arch Moore, served three terms as governor but also served time in prison for corruption.
Rockefeller was first elected to the Senate in 1984 after two terms as governor. He previously had served as secretary of state, a state lawmaker and West Virginia Wesleyan College's president. The great-grandson of oil tycoon John D. Rockefeller first arrived in the state in 1964 as a VISTA worker.