One paper, the Daily News, said Torre is being given an opportunity to resign first.
Saturday's 8-3 loss to the Detroit Tigers and elimination from the first round of the American League playoffs was a stunning loss for the highest payroll in Major League Baseball. And it was the second straight year New York lost in the opening round.
"It's certainly disappointing," Torre said after the game in Detroit. "Everyone in that locker room is disappointed."
The Yankees have won the World Series four times under Torre, most recently in 2000.
They had a record $200 million payroll this year and matched the New York Mets for the best record in the regular season (97-65). Detroit's payroll, by contrast, is less than half as much.
Nearly half of the top ten highest-paid players in baseball — including the top three, Alex Rodriguez, Derek Jeter and Jason Giambi — wear pinstripes. The highest paid Detroit Tiger, Magglio Ordonez, comes in at number 10.
Rodriguez has also come in for his share of the blame, hitting 1-for-14 against the Tigers, and 2-for-15 against the California Angels last year. Many fans and sportswriters say he should be among those sent packing from The Bronx.
After three years and half a billion dollars in player salaries without a World Series appearance, Steinbrenner — "The Boss — may be fed up, reports CBS News correspondent Bianca Solorzano.
He issued a statement that "this sad failure" was "absolutely not acceptable"
"It seems like the great job he was doing all year, all that's forgotten," former Yankees and Oakland Athletics star Reggie Jackson told The New York Times. "I imagine you could blame a guy for making bad moves, but I don't know how you can blame a guy for the team going 20 innings in a row without scoring a run."
In Torre's 11 years as manager, the Yankees have won more than 60 percent of their regular-season games (a 1079-and-699 record), with his 1,000th win as Yankees manager this past season.
The Yankees have made the playoffs in all 11 years that Torre has been their manager. They have won nine straight AL East titles, but that may not be enough.
"I think his message has been lost somehow," sportswriter Jon Heyman told CBS News. "For whatever reason he didn't get through to the players, they didn't perform to their abilities, and that reflects on the manager, and that's when it is time for a change."
The 66-year-old Torre has one year and $7 million left on his contract. He is baseball's highest-paid manager.
"Nobody gets to be Yankees manager forever, not even the great Joe Torre," wrote Daily News columnist Mike Lupica.
But William C. Rhoden in The New York Times said firing Torre was "unthinkable. The Yankees with Torre have been great for baseball for the last decade," Rhoden wrote. "He has earned the right to leave on his own terms."
Piniella is a former Yankees star and managed them in 1986-87 and for most of 1988. He guided Cincinnati to the 1990 World Series title and later managed Seattle and Tampa Bay.
The 63-year-old Piniella took a year off from managing last season, and is working as a broadcaster for the American League playoffs.
If he does replace Torre, "A-Rod" (Alex Rodriguez) could be a principal beneficiary: Piniella was his first major league manager, for seven years in Seattle, and they had a good relationship.