Murphy, 30, has until Nov. 13 to accept or reject the Mets' offer.
The decision ensures the Mets will recoup a draft pick after the first round if Murphy signs elsewhere this offseason.
Mets officials portray it as very likely that Murphy has played his final game as a Met.
Provided the team offsets the expected loss of free agent Yoenis Cespedes by bringing in a capable offensive outfielder, the Mets are prepared to use prospect Dilson Herrera at second base next season. Herrera, 21, would join a middle-infield combination with Wilmer Flores and Ruben Tejada.
The Mets might have been emboldened to make the qualifying offer because Murphy's postseason production could have increased the offers he will receive as a free agent. Murphy set a major league record with homers in six straight postseason games and earned National League Championship Series MVP honors.
Murphy did commit two errors during the World Series. He is capable of playing first base, second base and third base, and it will be interesting to see how different suitors view him. Third base is his natural position, and he stepped in there this past season when David Wright spent four months on the disabled list.
Murphy hit .281 with 38 doubles, 14 homers and 73 RBIs in 499 regular-season at-bats. The long ball total represented a career high as Murphy placed more emphasis on pulling the ball at the request of Mets hitting coach Kevin Long. Murphy also was the hardest to strike out among major league qualifiers, fanning only once every 14.2 plate appearances.
After Wright, Murphy is the second-longest-tenured Met. He was drafted by the organization in the 13th round in 2006 out of Jacksonville University and made his major league debut on Aug. 2, 2008.