Paxton (1-2) started the game, allowing three runs over five innings while allowing one hit and walking four, striking out eight. He threw 83 pitches.
"For Pax right now hoping for the best, that we get good news there and hopefully to continue to work to get him right," Boone said.
Shortstop Gleyber Torres also will undergo an MRI after leaving Thursday's game with a tight left hamstring, Boone said.
Torres stumbled while running down the first-base line after hitting a grounder in the bottom of the third inning. He was replaced in the top of the fourth by Thairo Estrada, who played second base while Tyler Wade shifted to shortstop.
Torres is batting .231 with one home run and six RBIs this season.
The injuries are the latest for high-profile Yankees players. Earlier Thursday, reliever Zack Britton was placed on the 10-day injured list because of a strained left hamstring.
He joined sluggers Giancarlo Stanton (hamstring) and Aaron Judge (calf) and infielder DJ LeMahieu (thumb), among impact players on the team to be placed on the 10-day injured list. Judge is tied for second in the American League with nine home runs, while LeMahieu leads the AL in hitting with a .411 average.
The Yankees were swept at home by the Rays, who took over first place in the AL East.