Rodriguez sat out Sunday night's game against the Baltimore Orioles after playing in the first half of the split-doubleheader. He was poised to meet the Yankees in Boston on Tuesday for the start of a three-game series.
"He's day to day," Girardi said. "I don't know if he'll be able to play (Tuesday). His thumb, we got to let it heal, so I can't tell you exactly when he's going to be back. It's just when he feels better."
Asked if A-Rod would be available at all in Boston, Girardi replied, "I sure hope so. It just depends on how the thumb responds to treatment. He came back and he seemed to irritate it a little bit more, we just have to be smart about it. I think it bothers him probably more hitting."
Shortstop Derek Jeter was out of the starting lineup Monday against the Orioles for a second straight game. He fouled a ball of his right knee in the third inning of the opener Sunday and sustained a bruised kneecap.
The knee swelled up between games, and although it was better Monday, Jeter was not ready to play a full nine innings.
"He was walking around fine today. He was walking a lot better than I would have anticipated when I saw him walk out last night," Girardi said. "He healed up pretty good. I can use him in case of an emergency."
Jeter played in his 2,402nd game with New York on Sunday, most in Yankees history. He moved one past Mickey Mantle.
Jeter didn't make it 2,403 on Monday but he was hoping to get back in the lineup on Tuesday against the Red Sox.
"That's the plan. I hope so," he said after New York's 3-2 win. "It feels a lot better than it did yesterday."
Jeter was pleased with the progress over a 24-hour period, noting that the knee was no longer swollen.
"Today, it just feels like I hit a ball off my knee," he said. "Yesterday, I don't know how to describe what happened."
New York trails first-place Boston by 1½ games, and Jeter intends to help the Yankees cut the deficit.
"We need to play well," he said. "They have a great team. Everyone knows that. It's a challenge for us. They've been playing extremely well so we need to be up for the challenge."
The Associated Press contributed to this story.