As the second round of American League balloting was released on Tuesday, Betts ranked fourth in the outfield with 519,146 votes, but is within striking distance of Michael Brantley (554,620) for the third spot.
"He contributes on all sides of the baseball for us," Red Sox manager John Farrell said of Betts.
Red Sox rookie outfielder Andrew Benintendi is also in the mix for a starting spot, as he currently ranks sixth with 416,461 votes.
Benintendi ripped two homers in Sunday's win over the Orioles, a sign that he might be ready to get hot again.