Rangers general manager Jon Daniels told The Dallas Morning News on Friday that "lines of communication remain open" between his club and Napoli.
Now, team No. 3 has a chance to make its pitch, according to CBS Sports.
Boston has a huge hole at first base after trading Adrian Gonzalez to Los Angeles in its house-cleaning blockbuster, and Napoli met with Red Sox brass this weekend, according to the Boston Globe.
Napoli, 31, has a combined 54 home runs the past two seasons, and he can play catcher, first base or serve as a designated hitter. Whichever team signs him will likely look at him as a hybrid rather than an anchor at one of those positions.
The sticking point in negotiations could be the difference between a three- and four-year deal, according to ESPN. Napoli's done great at Fenway Park in 24 career games, including the postseason. He has a .307/.404/.733 line with nine home runs and 20 RBIs in 75 at-bats.
The Mariners, meanwhile, want a middle-of-the-order bat in an otherwise young lineup.