The team has scheduled a news conference for Monday morning to make the announcement, the report said.
ESPN said La Russa received a multiyear contract. La Russa refused to confirm or deny the report.
"We're getting closer," the 65-year-old La Russa said Sunday. "I can't tell you anything yet."
The report also said the team has fired Hal McRae has hitting coach and will replace him with former Cardinals' slugger Mark McGwire.
ESPN said La Russa's contract is for one year plus a mutual option for 2011.
La Russa has managed the Cardinals for 14 years. He led the team to the National League Central Division title this season before they were swept by the Los Angeles Dodgers in the Division Series in three games.
His 2,552 career victories is third among managers all-time, behind Connie Mack (3,731) and John McGraw (2,763).
Astronomers offer more expansive view of universe
LGBT community has 'bullied the American people': Bachmann