Thirteen free agents received the offers, up from nine last year. The Yankees also extended offers to a trio of players: second baseman Robinson Cano, outfielder Curtis Granderson and pitcher Hiroki Kuroda.
Players accepting are signed for next season. If a player signs elsewhere, his former team gets an extra draft pick at the end of the first round next June as compensation.
Others receiving the offers were Atlanta catcher Brian McCann, Cincinnati outfielder Shin-Soo Choo, Cleveland pitcher Ubaldo Jimenez, Kansas City pitcher Ervin Santana, St. Louis outfielder Carlos Beltran, Seattle designated hitter Kendrys Morales and Texas outfielder Nelson Cruz.
There were 168 players who became major league free agents this year, up three from 2012. They were able to start talking contract with all teams on Tuesday.
The amount of the qualifying offer, which increased by $800,000 this year, is set by baseball's collective bargaining agreement as the average of the 125 highest contracts.
Among the nine players who received the offers last year, the only ones to stay with their teams were Kuroda, Red Sox designated hitter David Ortiz and Washington first baseman Adam LaRoche.
If a different club signs a player who received a qualifying offer, that team gives up a high draft pick. The top 10 overall selections in the draft cannot be forfeited, and a team signing multiple qualified players would lose a corresponding amount of selections.
Boston general manager Ben Cherington said the Red Sox decided not to make an offer to Jarrod Saltalamacchia, though he would like to re-sign the catcher.
"There's interest in every one of them," Cherington said of his free agents. "I also think it's unlikely that every one of them will be back. ... We're going to keep the conversation going with all of them, and also with alternatives, and see where the market shapes up. In a vacuum we'd like to have all of them back."
Cruz served a 50-game suspension this year; he was among 14 players disciplined by Major League Baseball after its investigation of the Biogenesis anti-aging clinic. In 108 games before his suspension in August, Cruz hit .269 with 27 homers and 76 RBIs. He returned for the AL wild-card tiebreaker and went 0 for 4.
Among those who did not receive offers were pitchers Bronson Arroyo of Cincinnati, Matt Garza of Texas, Roy Halladay of Philadelphia, Tim Hudson of Atlanta, Josh Johnson of Toronto and Fernando Rodney of Tampa Bay. Garza was ineligible because he wasn't with the Rangers on opening day.
A day after Tampa Bay exercised a $6.5 million option on David DeJesus, the sides reached an agreement on a two-year contract with a 2016 option. The deal is subject to a physical.
But the Rays declined a $2 million option on Juan Oviedo, the reliever who used to be known as Leo Nunez.
Colorado exercised a $4.25 million mutual option with reliever Matt Belisle.
San Francisco declined a $6.5 million option on right-hander Ryan Vogelsong, who gets a $300,000 buyout, and Baltimore turned down a $3 million option on infielder Alexi Casilla, who gets a $200,000 buyout. Atlanta declined a $1.6 million option on outfielder Reed Johnson, who receives a $150,000 buyout.
AP Sports Writer Jimmy Golen in Boston contributed to this report.