To make room for Lugo, sidelined since spring training with a knee injury, the Red Sox returned right-hander Michael Bowden to Triple-A Pawtucket.
Drew rested a sore left quadriceps. He left a 4-1 win over the New York Yankees on Sunday night.
"It feels better," Drew said. "I didn't know if it was cramps, but it felt tight and I told them that if I had to make a long run in the outfield or go first to third, there might be a problem."
Lugo was hurt in a March 13 exhibition game against the Yankees and opened the season on the 15-day disabled list after having arthroscopic surgery to repair a torn meniscus in his right knee.
"I feel good, but it is almost like spring training again for me," said Lugo, who had a four-game rehab stint at Pawtucket. "I'm eager to play."
Manager Terry Francona said Lugo and Drew would probably be in the lineup sometime during a three-game series against the Indians in Cleveland. Francona also gave an encouraging report on right-hander Daisuke Matsuzaka, on the disabled list since April 15 with a sore right shoulder.
"We activated Lugo, so he's ready," Francona said. "But he's had only 12 at-bats, so we'll be cautious and pay attention to how he's feeling.
"With J.D., we got him out of there so that it wasn't a multiple-week thing instead of one night. He's available. Dice-K was out to 120 feet [throwing] today and will do a bullpen session on Wednesday," Francona said.
Francona said that if that workout went well, Matsuzaka could soon start a rehab assignment in the minors.
Bowden was called up before Sunday's game against the Yankees and pitched two perfect innings in Boston's 4-1 victory.