"As of now he is the only one," Boone said.
Earlier this month, star slugger Aaron Judge was among six Yankees players who tested positive for coronavirus.
"Definitely, it's tough. It's been a tough year. We got this news with Cole and definitely a low blow there," Yankees catcher Gary Sanchez said through an interpreter.
Cole is 10-6 with a 3.11 ERA in 21 starts. The four-time All-Star is coming off a 14-0 loss at Tampa Bay on July 29 in which he allowed eight runs in 5⅓ innings.
The 30-year-old Cole is in his second season with the Yankees.
Boone said Nestor Cortes Jr. likely will start in place of Cole against the Orioles. Cortes has no record and a 1.93 ERA in three starts and eight relief appearances for the Yankees this season.
The Yankees' first game after the All-Star break on July 15 against Boston was postponed because of their virus outbreak.
Several teams around the majors have been affected recently by positive tests for the virus. Among them, the Milwaukee Brewers have put closer Josh Hader, star outfielder Christian Yelich and pitchers Jake Cousins, Jandel Gustave and Hunter Strickland on the COVID-19 injured list in the past week.